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, Guadalupe Gómez Melis 1 1 Universitat Politècnica de Catalunya 2 Intensive Care Unit, Donostia University Hospital, Donostia-San Sebastián March 6th, 2018

2 Outline Background Materials Methods Application Future work References

3 Acute Pancreatitis (AP) What is Acute Pancreatitis? An inflammatory condition of the pancreas characterized by symptoms of strong abdominal pain, nausea, vomiting, increased heart rate, fever and swollen and tender abdomen 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. 1/23

4 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. 2/23

5 Objectives Clinical goal Study of the clinical evolution of non-mild AP patients that enter to ICU. Methodological goal Apply multi-state models to obtain risk factors to help for an adequate prognosis of AP. Objective of the talk Describe the clinical problem, introduce the statistical methodology and show preliminary results. 3/23

6 Motivating dataset A prospective observational study. Carried out in Donostia University Hospital (Gipuzkoa). Patients (N = 286) that entered to the Intensive Care Unit with a diagnosis of Acute Pancreatitis. Follow-up: until their complete recovery (no longer than 31 Aug, 2017). Two treatment protocols conducted: one between 2001 and 2007 (93) and other from 2008 to mid-2017 (193). 4/23

7 Complete process for Acute Pancreatitis 5/23

8 Classical Survival data study Multi-state modeling In contrast to the classical statistical methods for analyzing survival data, multi-state models present many advantages: Study of multiple events of interest. The influence of the time at which intermediate events occur. Influence of prognostic factors on each of the intermediate event. Account for competing risks, unbiased estimates. Potential to obtain dynamic predictions. Predictions of the clinical prognosis of a patient at a certain point in his/her illness course. 6/23

9 The multi-state model 7/23

10 The multi-state model 7/23

11 The multi-state model 7/23

12 The multi-state model Notation: X = {X (t) : t 0, X (0) = 1} multi-state process. S = {1, 2, 3, 4} = {ICU, Hospital, Death, Home} state space. F t = σ{x (s) : s t} filtration of the process. 7/23

13 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 4 4 matrix for s t, (ii) Transition intensities, P(s, t) = {P ij (s, t; F s ); i, j S = {1, 2, 3, 4}} (2) 1 α ij (t; F t ) = lim t 0 + t P ij(t, t + t; F t ). (3) (iii) Cumulative (integrated) transition intensities, A ij (t; F t ) = t 0 α ij (u; F u ). (4) 8/23

14 Characterization of the multi-state model Assuming 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 4 4 matrix for s t, (ii) Transition intensities, P(s, t) = {P ij (s, t); i, j S = {1, 2, 3, 4}} (2) 1 α ij (t) = lim t 0 + t P ij(t, t + t). (3) (iii) Cumulative (integrated) transition intensities, A ij (t) = t 0 α ij (u). (4) 8/23

15 Characterization of the multi-state model Examples: (i) P 14 (0, 21) = P(X 21 = 4 X 0 = 1). Represents the probability that a patient is completely recovered after 3 weeks given that at time 0 she/he was in ICU. (ii) α 13 (5) = 1 lim t 0 + t P 13(5, 5 + t). Instantaneous risk of experiencing the 1 3 transition at time 5. Rate of dying in ICU at the 5th day. (iii) A 13 (5) = 5 0 α 13(u). Cumulative risk of dying in ICU by time 5. 8/23

16 Transition probability matrix Assuming Markov property, 1 P 12 (s, t) P 13 (s, t) P 14 (s, t) P 12 (s, t) P 13 (s, t) P 14 (s, t) P(s, t) = 0 1 P 23 (s, t) P 24 (s, t) P 23 (s, t) P 24 (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] 9/23

17 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 10/23

18 Overall survival time n Events Median 95% CI period: period: Survival functions according to treatment protocol Treatment S^(t) Time to death [days] 11/23

19 Model features Model with 4 states and 4 transitions. All the patients follow-up start at ICU, i.e. X (0) = 1. Overall 5 possible trajectories (see next slide). ICU and Hospital states are transient, Death and Home absorbing. We assume Markov property. We fit a non-parametric model, here. 12/23

20 All the possible paths 13/23

21 Multi-state modeling: observed transitions All the possible trajectories, to From ICU Hospital Death Home no event total entering ICU (80.4) 56 (19.5) Hospital (4.8) 219 (95.2) Death (100) 67 Home (100) 219 Table: Number of patients experiencing each transition during the follow-up. Frequencies and Proportions 1 1 Via paths() function from mstate R package 14/23

22 Multi-state modeling: duration of stay Group Group State N Min. Max. Median Mean N Min. Max. Median Mean ICU Hospital Table: Duration of stay (in days) in the two transient states 15/23

23 MS modeling: cumulative transition intensities period protocol period protocol ICU > Hospital ICU > Death Hospital > Death Hospital > Home ICU > Hospital ICU > Death Hospital > Death Hospital > Home Cumulative hazard Cumulative hazard Days from ICU entrance Days from ICU entrance Figure: Estimated cumulative transition intensities, Âij, per each i j transition. Left first period protocol group, right second period protocol group. 16/23

24 Multi-state modeling: transition probabilities Fixing a starting time of s= period protocol period protocol Home Probability Probability Home 0.2 Death 0.2 Death Hospital 0.0 ICU Days from ICU entrance Days from ICU entrance Hospital ICU 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, 4 and t 0. Left first period protocol group, right second period protocol group. 17/23

25 Multi-state modeling: transition probabilities Fixing a starting time of s=0.5 and t=21 days, 3 weeks period protocol period protocol Home Probability Probability Home 0.2 Death 0.2 Death Hospital 0.0 ICU Days from ICU entrance Days from ICU entrance Hospital ICU 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, 4 and t 0. Left first period protocol group, right second period protocol group. 17/23

26 Multi-state modeling: transition probabilities period protocol period protocol I H D Rec. I H ˆP(0.5, 21) = D Rec I H D Rec. I H ˆP(0.5, 21) = D Rec I H D Rec. I H s.e. : D Rec I H D Rec. I H s.e. : D Rec Via msfit() and probtrans() functions from mstate R package 18/23

27 Multi-state modeling: transition probabilities Another representation, period protocol period protocol ICU Hospital Death Home ICU Hospital Death Home Probability Probability Days from ICU entrance 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, 4 and t 0. Left first period protocol group, right second period protocol group. 19/23

28 Future work Fit a model to: determine those risk factors that have a stronger influence in the course of patients with AP. Prediction. Develop a predictive process to obtain predictions of the clinical prognosis. Build up a shiny app. 20/23

29 Future work: fitting Semi-parametric approach, Cox models: (Semi-)Markov stratified hazards (Semi-)Markov proportional hazards ICU Hospital α 12 (t Z 12 ) = α 12,0 (t) exp(β T Z 12 ) α 12 (t Z 12 ) = α 12,0 (t) exp(β T Z 12 ) ICU Death α 13 (t Z 13 ) = α 13,0 (t) exp(β T Z 13 ) α 13 (t Z 13 ) = α 3,0 (t) exp(β T Z 13 ) Hospital Death α 23 (t Z 23 ) = α 23,0 (t) exp(β T Z 23 ) α 23 (t Z 23 ) = α 3,0 (t) exp(β T Z 23 ) Hospital Home α 24 (t Z 24 ) = α 24,0 (t) exp(β T Z 24 ) α 24 (t Z 24 ) = α 24,0 (t) exp(β T Z 24 ) Covariates: period, age, bmi, gender, etiology, surgery, mechanical ventilation. Global covariates. Single effect estimate assumed to be common for all transitions. Transition-specific. Different covariate effect estimates accross all transition. 21/23

30 Future work: prediction Based on the best fit: estimate the conditional probability of some clinical feature event, given an history, and a set of values for prognostic factors Z of a patient. Given an alcoholic 67 years-old male patient who is in ICU since 10 days, what is the probability of being discharged 10 days later, i.e. after 20 days from admission? How does this probability compare to a baseline patient? 22/23

31 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), /23

32 Moltes gràcies!

33 Independent censoring? Watch out! Those censored individuals at time t are NOT representative for those still at risk at that time.

34 We should consider another state (Home): competing event of total recovery. A subject that is censored because of failure from a competing event risk, will with certainty NOT experience the event of interest.

35 Kaplan-Meier BIASED! It takes competing events as censored. Example The event of interest: the onset of a disease. Interested in: estimate F (t). Obviously, individuals may die without getting the disease. KM could consider death without the disease as independent censoring. An hypothetical population where individuals could not never die without the disease.

36 Make inference for disease risks and rates acknowledging the presence of the competing risks group group 1.0 Death 1.0 Death Probability Discharge Probability Discharge Days from ICU admission Days from ICU admission Figure: Cumulative incidence estimates of death and discharge, shown as survival and incidence curves, respectively; naïve Kaplan-Meier estimates are shown in grey