The Natural History of MELD

1. The Natural History of MELD Gordon Hazen INFORMS Healthcare June 21, 2011

2. 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

3. A MELD Progression Curiosity • UNOS MELD Data 2007

4. A MELD Progression Curiosity • Transition probabilities • Question: If not transplanted, does a patient tend to get better, or worse?

5. A MELD Progression Curiosity • For MELDs 21-30, and 15-20, the tendency is to improve if not transplanted: • 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.

6. Overview • Why this matters • So what can be done about this? • Natural history model • EM estimation • Results • Natural history • Naïve versus natural history • Summary

7. 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

8. Why this matters: Regional DA modeling • The naïve approach: • Keep (naïve) estimates of untransplanted MELD progression

9. 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.

10. 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.

11. 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

12. 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,1τrxy; all y) and total observation count (#Tx)rx++(#NoTx)rx+.

13. So what can be done? • Would like to form the maximum likelihood estimates • 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:

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

15. Results: Natural history • The E-M estimates of pxy(natural history) • Bold denotes a number larger than the corresponding naïve untransplanted progression probability. • Red denotes a number smaller than the corresponding naïve untransplanted progression probability.

16. Results: Naïve vs. E-M Natural history • MELD improvements for MELDs 21-30 and 15-20 are nearly eliminated.

17. DMELD: Naïve vs. E-M natural history • Using the following MELD assignments • the expected monthly change in MELD is:

18. Expected MELD progression: Naïve vs. E-M natural history

19. Untransplanted Progression • Note: Untransplanted progression = naïve progression  Natural history progression (the point of this talk)

20. 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:

21. News Flash: 12-month data • MELD improvements for MELDs 21-30 and 15-20 • January 2007 only:

22. News Flash: 12-month data • MELD improvements for MELDs 21-30 and 15-20 • 12-month data 2007:

23. 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?