A multi-state model for the prognosis of non-mild acute pancreatitis

Transcription

1 A multi-state model for the prognosis of non-mild acute pancreatitis Lore Zumeta Olaskoaga 1, Felix Zubia Olaskoaga 2, Marta Bofill Roig 1, Guadalupe Gómez Melis 1 1 Universitat Politècnica de Catalunya 2 Intensive Care Unit, Donostia University Hospital, Donostia-San Sebastián January 18th, 2018

2 Outline Background Materials Methods Application Future work References

3 Acute Pancreatitis (AP) What is Acute Pancreatitis? It is an inflammatory condition of the pancreas characterized by symptoms of strong abdominal pain, nausea, vomiting, increased heart rate, fever and swollen and tender abdomen It is a reversible process, most people with AP recover completely. It may range from mild discomfort to a severe, life-threatening illness. The major causes are biliary lithiasis and alcohol consumption. 1/14

4 Motivation and objectives Motivation The most severe forms, and consequently patients with AP admitted to Intensive Care Unit (ICU), showed high mortality. Prediction of AP mortality is not straightforward due to the low incidence of the most severe forms and because its fluctuating clinical course. Objectives Apply survival analysis, multi-state models, to try to achieve a better knowledge of the non-mild AP patients clinical evolution. 2/14

5 Survival analysis background What is survival analysis? It is a collection of statistical procedures for data analysis, for which the outcome variable of interest is a positive random variable such as T time until an event D occurs. A peculiarity: Censoring We do not observe all the times Censored times. The distribution of T is not symmetric The normal model is not appropriate. So a specific methodology is needed. 3/14

6 Motivating dataset A prospective observational study. Carried out in Donostia University Hospital (Gipuzkoa). Patients that entered to the Intensive Care Unit with a diagnosis of Acute Pancreatitis. Follow-up: until their complete recovery. Two treatment protocols conducted: one between 2001 and 2007 and other from 2008 to mid N=286 patients and several measurements taken. 4/14

7 Complete process for Acute Pancreatitis 5/14

8 The multi-state illness-death model Notation: X = {X (t) : t 0, X (0) = 1} multi-state process. S = {1, 2, 3} = {ICU, Hospital, Death due to AP} state space. F t = σ{x (s) : s t} filtration of the process. 6/14

9 Characterization of the multi-state model The law of a multi-state process is defined by its finite dimensional distribution and fully characterized through either one of the following functions: (i) Transition probability between state i and state j for s t, P ij (s, t; F s ) = P(X t = j X s = i; F s ). (1) Transition probability k k matrix for s t, P(s, t) = {P ij (s, t; F s ); i, j S = {0, 1,..., k}} (2) (ii) Transition intensities, α ij (t; F t ) = (iii) Cumulative (integrated) transition intensities, 1 lim t 0 + t P ij(t, t + t; F s ). (3) A ij (t; F t ) = t 0 α ij (u; F u ). (4) 7/14

10 Characterization of the multi-state model (i) Transition probability between state i and state j for s t, P ij (s, t; F s ) = P(X t = j X s = i; F s ). (1) Transition probability k k matrix for s t, P(s, t) = {P ij (s, t; F s ); i, j S = {0, 1,..., k}} (2) (ii) Transition intensities, α ij (t; F t ) = (iii) Cumulative (integrated) transition intensities, 1 lim t 0 + t P ij(t, t + t; F s ). (3) A ij (t; F t ) = t 0 α ij (u; F u ). (4) 7/14

11 Characterization of the multi-state model Due to Markov property: (i) Transition probability between state i and state j for s t, P ij (s, t) = P(X t = j X s = i). (1) Transition probability 3 3 matrix for s t, (ii) Transition intensities, P(s, t) = {P ij (s, t); i, j S = {1, 2, 3}} (2) α ij (t) = (iii) Cumulative (integrated) transition intensities, 1 lim t 0 + t P ij(t, t + t). (3) A ij (t) = t 0 α ij (u). (4) 7/14

12 Transition probability matrix Due to the Markov property of this particular model, 1 P 12 (s, t) P 13 (s, t) P 12 (s, t) P 13 (s, t) P(s, t) = 0 1 P 23 (s, t) P 23 (s, t), and this can be recovered from the transition intensities, A(t), through product integration P(s, t) = (I + da(u)). u (s,t] 8/14

13 Transition probability matrix: estimation To obtain ˆP(s, t) it is enough to compute the Aalen-Johansen estimator of the cumulative transition intensity da: dâij(t) = dn ij(t) Y i (t),  ij (t) = u t dâij(u), i j and  ii (t) = i j dâij(t). Being, N ij (t) = number of observed direct transitions from state i to state j up to time t Y i (t) = number of individuals under observation in state i just before time t da(t): matrix of elements d(a ij (t)) i,j = (α ij (t)) i,j dt 9/14

14 Multi-state modeling: transition matrix All the possible trajectories, to From ICU Hospital Death no event total entering ICU (80.4) 56 (19.5) (100) Hospital (4.8) 219 (95.2) 230 (100) Death (100) 67 (100) Table: Number of patients experiencing each transition during the follow-up. Frequencies and Proportions 1 1 Via paths() function from mstate R package 10/14

15 Multi-state modeling: transition probabilities Fixing a starting time of s= period protocol period protocol Death 0.8 Death 0.8 Probability Probability Hospital 0.2 Hospital ICU Days from ICU entrance 0.0 ICU Days from ICU entrance Figure: Transition probabilities, starting in state 1 (ICU) at time 0.5, i.e. P 1j (0.5, t), for all j = 1, 2, 3 and t 0. Left first period protocol group, right second period protocol group. 11/14

16 Multi-state modeling: transition probabilities Fixing a starting time of s=0.5 and t=21 days, 3 weeks period protocol period protocol Death 0.8 Death 0.8 Probability Probability Hospital 0.2 Hospital ICU Days from ICU entrance 0.0 ICU Days from ICU entrance Figure: Transition probabilities, starting in state 1 (ICU) at time 0.5, i.e. P 1j (0.5, t) for all j = 1, 2, 3 and t 0. Left first period protocol group, right second period protocol group. 11/14

17 Multi-state modeling: transition probabilities P 1 (0.5, 21) = P 2 (0.5, 21) = Via msfit() and probtrans() functions from mstate R package 12/14

18 Future work Risk factors: model the effect of covariates on transition i j using Cox s proportional hazards model on the α ij (t Z) transition hazards. Evaluate the relationship between surgery and mortality. Develop a predictive process that allows the risk of a patient to be updated whenever new information of his or her evolution is available. 13/14

19 References Di M-Y., Liu H., Yang Z-Y., Bonis P. A., Tang J-L, Lau J.(2016) Prediction Models of Mortality in Acute Pancreatitis in Adults: A Systematic Review. Annals of Internal Medicine. 165(7): Forsmark C. E., Yadav D. (2016) Predicting the Prognosis of Acute Pancreatitis. Annals of Internal Medicine. 165(7): Meira-Machado L., Uña-Álvarez, J. De, Cadarso-Suárez, C., Andersen P. K. (2009). A multi-state model for the analysis of time-to-event data Stat. Methods Med Res. Vol. 18(2), Hougaard P. (2000) Analysis of multivariate survival data, Springer, Chapter 5. Porta N., Calle M. Luz, Malats N. and Gómez L. (2012). A dynamic model for the risk of bladder cancer progression. Statistics in Medicine. Vol. 31(3), Wreede L. C. de, Fiocco M. and Putter H. (2011).mstate: An R Package for the Analysis of Competing Risks and Multi-State Models. Journal of Statistical Software. Vol. 38(7), /14

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