THE NATURAL HISTORY OF MELDGordon HazenINFORMS Healthcare June 21, 2011

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MELD The U.S. liver transplant wait list is

prioritized by MELD.MELD = Model for End-Stage Liver DiseaseA combination of laboratory values

positively correlated with 90-day mortalityCox Regression:

MELD = 3.78[Ln serum bilirubin (mg/dL)] + 11.2[Ln INR] + 9.57[Ln serum creatinine (mg/dL)] + 6.43

Truncated to the range 6 – 40 Instituted by UNOS in 2002

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A MELD PROGRESSION CURIOSITY UNOS MELD Data 2007

30 day beginning Jan 1, 2007RemovedRemovedStill Listed

Transplanted

Died w/o Tx Other

MELD 31+

MELD 21 - 30

MELD 15 - 20

MELD 11 - 14 MELD <11

MELD 31+ 24 7 10 6 4 0 1 0 28MELD 21 - 30 83 15 19 11 220 59 6 1 331MELD 15 - 20 116 13 24 6 124 1,874 206 20 2267MELD 11 - 14 35 10 37 1 11 264 3,470 217 4010MELD < 11 28 7 54 0 4 19 209 4,740 5033

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A MELD PROGRESSION CURIOSITY Transition probabilities

RemovedRemovedStill ListedTransplanted

Died w/o Tx Other

MELD 31+

MELD 21 - 30

MELD 15 - 20

MELD 11 - 14 MELD <11

MELD 31+ 0.46 0.25 0.36 0.21 0.14 0.00 0.04 0.00MELD 21 - 30 0.20 0.05 0.06 0.03 0.66 0.18 0.02 0.00MELD 15 - 20 0.05 0.01 0.01 0.00 0.05 0.83 0.09 0.01MELD 11 - 14 0.01 0.00 0.01 0.00 0.00 0.07 0.87 0.05MELD < 11 0.01 0.00 0.01 0.00 0.00 0.00 0.04 0.94

Question: If not transplanted, does a patient tend to get better, or worse?

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A MELD PROGRESSION CURIOSITY For MELDs 21-30, and 15-20, the tendency is to

improve if not transplanted:Worse Better

MELD 31+ 0.61 0.18MELD 21 - 30 0.14 0.20MELD 15 - 20 0.07 0.10MELD 11 - 14 0.08 0.05MELD < 11 0.06

Worse Incl Tx Better0.79 0.100.31 0.160.12 0.090.09 0.050.06

Possible explanation: Transplant tends to censor worsening MELDs more than it censors improving MELDs.

Implication: We do not know the natural history of MELD progression.

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OVERVIEW Why this matters So what can be done about this?

Natural history modelEM estimation

ResultsNatural historyNaïve versus natural history

Summary

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WHY THIS MATTERS: REGIONAL DA MODELING Transplant rates differ across regions Therefore, decision analyses should be

done separately by regionUse regional transplant probabilitiesUse national MELD progression probabilities

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WHY THIS MATTERS: REGIONAL DA MODELING The naïve approach:

RemovedRemovedStill ListedTransplanted

Died w/o Tx Other

MELD 31+

MELD 21 - 30

MELD 15 - 20

MELD 11 - 14 MELD <11

MELD 31+ 0.46 0.25 0.36 0.21 0.14 0.00 0.04 0.00MELD 21 - 30 0.20 0.05 0.06 0.03 0.66 0.18 0.02 0.00MELD 15 - 20 0.05 0.01 0.01 0.00 0.05 0.83 0.09 0.01MELD 11 - 14 0.01 0.00 0.01 0.00 0.00 0.07 0.87 0.05MELD < 11 0.01 0.00 0.01 0.00 0.00 0.00 0.04 0.94

Region 10MELD 31+ 0.25MELD 21-30 0.48MELD 15-20 0.101MELD 11-14 0.034MELD <11 0.01

Region 1MELD 31+ 0.5MELD 21-30 0.077MELD 15-20 0MELD 11-14 0MELD <11 0.009

• Keep (naïve) estimates of untransplanted MELD progression

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WHY THIS MATTERS: REGIONAL DA MODELING If region has low transplant rates, then

Fewer bad MELD transitions are censored; so Untransplanted MELD progression should be

worse than the national average If region has high transplant rates, then

More bad MELD transitions are censored Untransplanted MELD progression should be

better than the national average The (naïve) national estimates of

untransplanted MELD progression do not reflect these changes.

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WHY THIS MATTERS: DA POLICY MODELING If a policy change lowers transplant rates,

then Fewer bad MELD transitions are censored; so Untransplanted MELD progression should be worse

than before If a policy change raises transplant rates, then

More bad MELD transitions are censored Untransplanted MELD progression should be

better than before The (naïve) national estimates of

untransplanted MELD progression do not reflect these changes.

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SO WHAT CAN BE DONE? Estimate natural history of MELD progression

pxy = transition prob from MELD category x to category y in the absence of any transplants

Estimate region-specific transplant probs trxy = prob in region r of transplant given MELD

transition from category x to category y The complete-data likelihood

(#Tx) (#NoTx)(1 )rxy rxy

c xy rxy xy rxyr x yL p pt t

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SO WHAT CAN BE DONE?

We see therefore that Lc is the product of (a) transition data: the product over x of independent

multinomial observations ((#Tx)+xy + (NoTx)+xy; all y)

with category probabilities (pxy; all y) and total observation count (#Tx)+x+ + (#NoTx)+x+ ; and

(b) transplant data: the product over r and x of independent multinomial observations

((#Tx)rxy , (#NoTx)rxy; all y) with category probabilities (τrxy,1τrxy; all y) and total observation count (#Tx)rx++(#NoTx)rx+.

(#Tx) (#NoTx)

(#Tx) (#NoTx) (#Tx) (#NoTx)

(1 )

(1 )

rxy rxy

xy xy rxy rxy

c xy rxy xy rxyr x y

xy rxy rxyx y r x y

L p p

p

t t

t t

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SO WHAT CAN BE DONE? Would like to form the maximum likelihood estimates

(# Tx)ˆ

(#Tx) (# Tx(#Tx)

)xy

xyx

xy

x

nop

no

(#Tx)(#T

ˆ(# Tx) x)rxy

rxr

yxy

rxynot

(#Tx) (#Tx) xy rxyrxy rx

xy rxyy

p

p

t

t

We do observe (#Tx)rx+. So if we knew pxy and trxy, we could calculate the expected value of the unobserved (#Tx)rxy:

But how to do this if we cannot observe (#Tx)rxy = # in region r who went from x to y and were transplanted?

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SO WHAT CAN BE DONE? This is a missing data problem, for which the E-M

algorithm is known to be a useful tool. The E-M algorithm:

ˆ ˆStart with some estimates , of , .

ˆ ˆ(E) Pretend , are , and calculate (#Tx) .

ˆ ˆ(M) Pretend (#Tx) is (#Tx) and form MLE estimates , .

Repeat until estimates c

xy rxy xy rxy

xy rxy xy rxy rxy

rxy rxy xy rxy

p p

p p

p

t t

t t

t

onverge.

The E-M algorithm is known to converge to at least a local MLE.

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RESULTS: NATURAL HISTORY The E-M estimates of pxy (natural history)

Died w/o Tx Other

MELD 31+

MELD 21 - 30

MELD 15 - 20

MELD 11 - 14

MELD <11

MELD 31+ 0.29 0.33 0.20 0.12 0.01 0.04 0.01MELD 21 - 30 0.07 0.08 0.04 0.60 0.18 0.02 0.00MELD 15 - 20 0.01 0.02 0.00 0.06 0.81 0.09 0.01MELD 11 - 14 0.00 0.01 0.00 0.00 0.07 0.86 0.06MELD < 11 0.00 0.01 0.00 0.00 0.00 0.04 0.94

• Bold denotes a number larger than the corresponding naïve untransplanted progression probability.

• Red denotes a number smaller than the corresponding naïve untransplanted progression probability.

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RESULTS: NAÏVE VS. E-M NATURAL HISTORY MELD improvements for MELDs 21-30 and 15-20

are nearly eliminated.NaïveWorse Better

MELD 31+ 0.61 0.18MELD 21 - 30 0.14 0.20MELD 15 - 20 0.07 0.10MELD 11 - 14 0.08 0.05MELD < 11 0.06

E-MWorse Better

0.62 0.170.19 0.200.09 0.100.09 0.060.06

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DMELD: NAÏVE VS. E-M NATURAL HISTORY Using the following MELD assignments

50 45 35.5 25.5 17.5 12.5 8Died w/o Tx

Other removal

MELD 31+

MELD 21 - 30

MELD 15 - 20

MELD 11 - 14

MELD <11

Naïve E-MMELD 31+ 4.556 4.939MELD 21 - 30 0.782 1.979MELD 15 - 20 0.335 0.648MELD 11 - 14 0.655 0.762MELD < 11 0.665 0.754

DMELD

• the expected monthly change in MELD is:

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EXPECTED MELD PROGRESSION: NAÏVE VS. E-M NATURAL HISTORY

0 10 20 300

10

20

30

40

50

31+21-3015-2011-14<11

Using naive estimates

Month

Expe

cted

MEL

D

0 10 20 300

10

20

30

40

50

31+21-3015-2011-14<11

Using E-M estimates

Month

Expe

cted

MEL

D

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UNTRANSPLANTED PROGRESSION Note: Untransplanted progression = naïve

progression Natural history progression (the point of this talk)

1

1

P( | )

Untransplanted progression P

from to in region P( | , Region = , No Tx)

(1 )(1 )

xy t t

rxy

t t

rxy xy

rxy xyy

p M y M x

qx y r

M y M x rpp

tt

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RESULTS: PROJECTED IMPACT OF DTRANSPLANT RATE ON (NAÏVE) UNTRANSPLANTED MELD PROGRESSION What happens if we scale up/down the transplant probabilities trxy? Do we

see the predicted change in naïve progression? For Region 7:

0 0.5 1 1.5 20

0.2

0.4

0.6

MELD 31+MELD 21-30MELD 15-20MELD 11-14MELD < 11

Multiple of regional tx rate

Mon

thly

pro

b M

ELD

up

0 0.5 1 1.5 20

0.2

0.4

MELD 31+MELD 21-30MELD 15-20MELD 11-14MELD < 11

Multiple of regional tx rate

Mon

thly

pro

b M

ELD

dow

n

0 0.5 1 1.5 21

0

1

2

3

4

5

MELD 31+MELD 21-30MELD 15-20MELD 11-14MELD < 11

Multiple of regional tx rate

Mon

thly

Del

ta M

ELD

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NEWS FLASH: 12-MONTH DATA MELD improvements for MELDs 21-30 and 15-20 January 2007 only:

NaïveWorse Better

MELD 31+ 0.61 0.18MELD 21 - 30 0.14 0.20MELD 15 - 20 0.07 0.10MELD 11 - 14 0.08 0.05MELD < 11 0.06

E-MWorse Better

0.62 0.170.19 0.200.09 0.100.09 0.060.06

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NEWS FLASH: 12-MONTH DATA MELD improvements for MELDs 21-30 and 15-20 12-month data 2007:

NaïveWorse Better

MELD 31+ 0.49 0.25MELD 21 - 30 0.06 0.22MELD 15 - 20 0.05 0.08MELD 11 - 14 0.06 0.04MELD < 11 0.04 -

EMWorse Better0.44 0.160.15 0.230.08 0.090.07 0.040.05 -

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SUMMARY E-M estimation can be used to capture

natural history of MELD. E-M estimates confirm that transplanting

censors worsening MELD progression more than it does improving MELD progression.

The difference is not large on a monthly basis but can compound to make a difference.

MELD 21-30 natural history estimates still indicate a tendency to improve – is something else going on?