THE NATURAL HISTORY OF
MELD
Gordon Hazen
INFORMS Healthcare June 21, 2011
MELD
 The
U.S. liver transplant wait list is
prioritized by MELD.



MELD = Model for End-Stage Liver Disease
A combination of laboratory values positively
correlated with 90-day mortality
Cox 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
2
A MELD PROGRESSION CURIOSITY

UNOS MELD Data 2007
30 day beginning Jan 1, 2007
RemovedRemovedStill Listed
Transpl Died
MELD MELD MELD MELD
anted w/o Tx Other 31+
21 - 30 15 - 20 11 - 14 MELD <11
MELD 31+
24
7
10
6
4
0
1
0
MELD 21 - 30
83
15
19
11
220
59
6
1
MELD 15 - 20
116
13
24
6
124 1,874
206
20
MELD 11 - 14
35
10
37
1
11
264
3,470
217
MELD < 11
28
7
54
0
4
19
209
4,740
28
331
2267
4010
5033
3
A MELD PROGRESSION CURIOSITY

Transition probabilities
RemovedRemovedStill Listed
Transpl Died
MELD MELD MELD MELD
anted w/o Tx Other 31+
21 - 30 15 - 20 11 - 14
MELD 31+
0.46
0.25
0.36
0.21
0.14
0.00
0.04
MELD 21 - 30
0.20
0.05
0.06
0.03
0.66
0.18
0.02
MELD 15 - 20
0.05
0.01
0.01
0.00
0.05
0.83
0.09
MELD 11 - 14
0.01
0.00
0.01
0.00
0.00
0.07
0.87
MELD < 11
0.01
0.00
0.01
0.00
0.00
0.00
0.04

MELD <11
0.00
0.00
0.01
0.05
0.94
Question: If not transplanted, does a patient tend to
get better, or worse?
4
A MELD PROGRESSION CURIOSITY

For MELDs 21-30, and 15-20, the tendency is to
improve if not transplanted:
MELD 31+
MELD 21 - 30
MELD 15 - 20
MELD 11 - 14
MELD < 11


Worse Better
0.61
0.18
0.14
0.20
0.07
0.10
0.08
0.05
0.06
Worse
Incl Tx Better
0.79
0.10
0.31
0.16
0.12
0.09
0.09
0.05
0.06
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.
5
OVERVIEW
 Why
this matters
 So what can be done about this?


Natural history model
EM estimation
 Results


Natural history
Naïve versus natural history
 Summary
6
WHY THIS MATTERS: REGIONAL DA
MODELING
 Transplant
rates differ across regions
 Therefore, decision analyses should be done
separately by region
Use regional transplant probabilities
 Use national MELD progression probabilities

7
WHY THIS MATTERS: REGIONAL DA
MODELING
 The
naïve approach:
RemovedRemovedStill Listed
Transpl Died
MELD MELD MELD MELD
anted w/o Tx Other 31+
21 - 30 15 - 20 11 - 14
MELD 31+
0.46
0.25
0.36
0.21
0.14
0.00
0.04
MELD 21 - 30
0.20
0.05
0.06
0.03
0.66
0.18
0.02
MELD 15 - 20
0.05
0.01
0.01
0.00
0.05
0.83
0.09
MELD 11 - 14
0.01
0.00
0.01
0.00
0.00
0.07
0.87
MELD < 11
0.01
0.00
0.01
0.00
0.00
0.00
0.04
Region 10
MELD 31+
0.25
MELD 21-30 0.48
MELD 15-20 0.101
MELD 11-14 0.034
MELD <11
0.01
Region 1
MELD 31+
0.5
MELD 21-30 0.077
MELD 15-20
0
MELD 11-14
0
MELD <11
0.009
MELD <11
0.00
0.00
0.01
0.05
0.94
• Keep (naïve)
estimates of
untransplanted 8
MELD progression
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.
9
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.
10
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
Lc 
   p
r
x
y
xyt rxy

(#Tx)rxy
 pxy (1  t rxy ) 
(#NoTx)rxy
11
SO WHAT CAN BE DONE?
   p
  p
Lc 
r
x

x
y
y
xyt rxy

(#Tx)rxy
(#Tx) xy  (#NoTx)  xy
xy
 pxy (1  t rxy ) 

r
x
(#NoTx) rxy
t rxy
(#Tx) rxy
y
(1  t rxy )
(#NoTx) rxy
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,1τrxy; all y) and total
observation count (#Tx)rx++(#NoTx)rx+.

12
SO WHAT CAN BE DONE?

Would like to form the maximum likelihood estimates
pˆ xy 


(# Tx)  xy  (# noTx)  xy
tˆrxy 
(# Tx)  x   (# noTx ) x 
(#Tx)rxy
(#Tx)rxy  (# noTx)rxy
But how to do this if we cannot observe (#Tx)rxy = # in
region r who went from x to y and were transplanted?
We do observe (#Tx)rx+. So if we knew pxy and trxy, we could
calculate the expected value of the unobserved (#Tx)rxy:
(# Tx) rxy  (# Tx) rx 
pxyt rxy

y
pxyt rxy
13
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:

Start with some estimates pˆ xy ,tˆrxy of pxy ,t rxy .
(E) Pretend pxy ,t rxy are pˆ xy ,tˆrxy and calculate (# Tx) rxy .
(M) Pretend (# Tx) rxy is (# Tx) rxy and form MLE estimates pˆ xy ,tˆrxy .
Repeat until estimates converge.

The E-M algorithm is known to converge to at least a
local MLE.
14
RESULTS: NATURAL HISTORY

The E-M estimates of pxy (natural history)
Died
w/o Tx Other
MELD 31+
0.29
0.33
MELD 21 - 30
0.07
0.08
MELD 15 - 20
0.01
0.02
MELD 11 - 14
0.00
0.01
MELD < 11
0.00
0.01
MELD MELD MELD MELD MELD
31+
21 - 30 15 - 20 11 - 14 <11
0.20
0.12
0.01
0.04
0.01
0.04
0.60
0.18
0.02
0.00
0.00
0.06
0.81
0.09
0.01
0.00
0.00
0.07
0.86
0.06
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.
15
RESULTS: NAÏVE VS. E-M NATURAL
HISTORY

MELD improvements for MELDs 21-30 and 15-20 are
nearly eliminated.
Naïve
Worse Better
MELD 31+
0.61
0.18
MELD 21 - 30
0.14
0.20
MELD 15 - 20
0.07
0.10
MELD 11 - 14
0.08
0.05
MELD < 11
0.06
E-M
Worse Better
0.62
0.17
0.19
0.20
0.09
0.10
0.09
0.06
0.06
16
DMELD: NAÏVE VS. E-M NATURAL
HISTORY

Using the following MELD assignments
50
45
35.5
Died
Other
MELD
w/o Tx removal 31+
25.5
17.5
12.5
8
MELD MELD MELD MELD
21 - 30 15 - 20 11 - 14 <11
• the expected monthly change in MELD is:
MELD 31+
MELD 21 - 30
MELD 15 - 20
MELD 11 - 14
MELD < 11
DMELD
Naïve
E-M
4.556
4.939
0.782
1.979
0.335
0.648
0.655
0.762
0.665
0.754
17
EXPECTED MELD PROGRESSION: NAÏVE
VS. E-M NATURAL HISTORY
Using E-M estimates
50
50
40
40
Expected MELD
Expected MELD
Using naive estimates
30
31+
21-30
15-20
11-14
<11
20
10
0
0
10
20
Month
30
31+
21-30
15-20
11-14
<11
20
10
30
0
0
10
20
Month
30
18
UNTRANSPLANTED PROGRESSION

Note: Untransplanted progression = naïve progression
 Natural history progression (the point of this talk)
pxy  P( M t 1  y | M t  x)
qrxy
 Untransplanted progression 
 P

 from x to y in region r

 P( M t 1  y | M t  x, Region = r , No Tx)

(1  t rxy ) pxy

(1

t
)
p
rxy
xy
y
19
RESULTS: PROJECTED IMPACT OF DTRANSPLANT RATE
ON (NAÏVE) UNTRANSPLANTED MELD PROGRESSION
Monthly prob MELD up
0.6
0.4
0.2
0
0
0.5
1
1.5
Multiple of regional tx rate
MELD 31+
MELD 21-30
MELD 15-20
MELD 11-14
MELD < 11
2
5
Monthly Delta MELD

What happens if we scale up/down the transplant probabilities
trxy? Do we see the predicted change in naïve progression?
For Region 7:
Monthly prob MELD down

0.4
0.2
4
3
2
1
0
1
0
0
0.5
1
1.5
Multiple of regional tx rate
MELD 31+
MELD 21-30
MELD 15-20
MELD 11-14
MELD < 11
2
0
0.5
1
1.5
2
Multiple of regional tx rate
MELD 31+
MELD 21-30
MELD 15-20
MELD 11-14
MELD < 11
20
NEWS FLASH: 12-MONTH DATA
MELD improvements for MELDs 21-30 and 15-20
 January 2007 only:

Naïve
Worse Better
MELD 31+
0.61
0.18
MELD 21 - 30
0.14
0.20
MELD 15 - 20
0.07
0.10
MELD 11 - 14
0.08
0.05
MELD < 11
0.06
E-M
Worse Better
0.62
0.17
0.19
0.20
0.09
0.10
0.09
0.06
0.06
21
NEWS FLASH: 12-MONTH DATA
MELD improvements for MELDs 21-30 and 15-20
 12-month data 2007:

Naïve
Worse
MELD 31+
0.49
MELD 21 - 30 0.06
MELD 15 - 20 0.05
MELD 11 - 14 0.06
MELD < 11
0.04
Better
0.25
0.22
0.08
0.04
-
EM
Worse
0.44
0.15
0.08
0.07
0.05
Better
0.16
0.23
0.09
0.04
22
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?

23