# Markov State Modelling of Disease Courses and Mortality Risks of Patients with Community-Acquired Pneumonia

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## Abstract

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## 1. Introduction

## 2. Materials and Method

#### 2.1. Patient Data

#### 2.2. Defining States of CAP Severity

#### 2.3. Establishing the Markov Model

#### 2.4. Comparisons with Other Risk Scores

## 3. Results

#### 3.1. Comparison of Model and Data

#### 3.2. Transition Probability Matrices

#### 3.3. Predicting 28 d Mortality

#### 3.4. Distribution of Sojourn Times

#### 3.5. Prediction of Death for Patients with Initial Severe CAP

#### 3.6. Clinical Utility of the Model

^{−4}, PSI: p = 6.2 × 10

^{−3}, CURB-65: p = 0.024).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### A.1. Details of Markov Modelling

**P**is the transition probability matrix,

**Q**the transition intensity matrix, and

**t**is the time. Each entry

**q**of the matrix

_{ij}**Q**corresponds to the transition rate from disease state

**i**to disease state

**j**. Since we have five disease states,

**Q**is a 5x5 matrix. The last row of

**Q**is zero because death is an absorbing state. Since we consider stepwise deteriorations and improvements of disease state (see Figure 2 in main document), the transition intensity matrix can be further simplified to the structure displayed in Equation (A2). The equation was solved numerically by a method that makes an expansion of the matrix exponential for the transition rates.

#### A.2. Transition Probabilities of SepNet Studies in Dependence on Number of Observation Days Used for Model-Calibration

**Figure A1.**Comparison of transition probabilities between disease states estimated for the SepNet trials in dependence on the number of data points used for calibration (5, 10, 15, and 20 days). Each subfigure shows the daily transition probabilities from one disease state (title of figure) to the other possible disease states shown on the x-axis. Transition probabilities corresponding to calibration on 15 days are also shown in Figure 5 of the main document. We added the transition probabilities of PROGRESS for comparison. Good agreements of transition probabilities between studies and calibration scenarios were observed. An exception is VISEP showing stronger dependence of estimates on the number of time points used for calibration.

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**Figure 2.**Markov model of community-acquired pneumonia (CAP) disease course with four disease states of increasing severity and death as absorbing state. Disease ameliorations and deteriorations occur stepwise, while death is possible at any state.

**Figure 3.**Results of Markov modelling (red curves) in comparison to data of the PROGRESS study (blue curves). Each subfigure shows the proportion of patients in the respective disease states over the first five days. The data fit well with the respective predictions of the Markov model.

**Figure 4.**Results of Markov modelling (dashed curves) in comparison to data of the SepNet studies (solid curves) characterized by longer time series. Each panel shows the proportion of patients in the respective disease state over the first 15 days. The filled circles represent model predictions for days not included in model calibration. The data of the three studies fit well with the respective predictions of the Markov model.

**Figure 5.**Comparison of transition probabilities between disease states across studies and possible state transitions. To estimate the probabilities, we used 5 observation days for PROGRESS and 15 observation days for all SepNet studies. Each panel shows the daily transition probabilities from one disease state (see panel title) to the other possible disease states shown on the x-axis. Good agreements of transition probabilities between studies were observed.

**Figure 6.**Comparison of mortality predicted by Markov modelling (dashed curves) in comparison to data of PROGRESS and the SepNet studies (solid curves). Agreement between model predictions and data not used for model calibration was excellent (filled circles). VISEP showed slightly inferior prediction.

**Figure 7.**Distribution of sojourn times for disease states. For SepNet studies, we used the transition matrix obtained from fitting the first 15 days. Shown are the median and the 25% and 75% quartile of calculated sojourn times.

**Figure 8.**Model validation regarding 28 d mortality was performed using the severely ill CAP patients (ssCAP sub-cohort) of PROGRESS. Predictions and data were in good agreement. Moreover, the model correctly predicted the increased mortality of ssCAP patients compared to PROGRESS patients with mostly lower initial CAP severity used for model calibration (compare with Figure 6).

**Figure 9.**ROC curves for 28 d mortality in the PROGRESS data using established scoring systems PSI (area under curve (AUC) = 0.78, 95% CI: 0.71–0.86) and CURB-65 (AUC = 0.84, 95% CI: 0.76–0.89) in comparison to our initial disease states (AUC = 0.76, 95% CI: 0.70–0.83) and our model-based risk assessment (AUC = 0.89, 95% CI 0.84–0.94).

**Table 1.**Study Characteristics. Age and initial Sequential Organ Failure Assessment (SOFA) were reported as medians and interquartile ranges. * Subcohort of PROGRESS patients without study visits.

Study | Age | Sex (m/f) | Initial SOFA | Type of Study | Maximum Observation Time (d) | 28d Mortality N (%) | Clinical trial.gov Identifier |
---|---|---|---|---|---|---|---|

PROGRESS | 62 (45,74) | 1081/782 | 2 (1,3) | Observational study | 5 | 37 (2.0%) | NCT02782013 |

ssCAP * | 69 (55,76) | 104/38 | 7 (6,8) | Observational study | 1 | 8 (5.6%) | NCT02782013 |

MAXSEP | 67 (57,74) | 152/56 | 10 (8,12) | Randomized trial | 22 | 39 (18.8%) | NCT00534287 |

VISEP | 66 (57,74) | 138/65 | 10 (7.5,12) | Randomized trial | 22 | 54 (26.6%) | NCT00135473 |

SISPCT | 67.5 (56,74) | 300/122 | 9 (7,12) | Randomized Trial | 22 | 97 (23.0%) | NCT00832039 |

Disease State | SOFA Score (SC) Range |
---|---|

S1 | 0 ≤ SC ≤ 2 |

S2 | 2 < SC ≤ 5 |

S3 | 5 < SC ≤ 9 |

S4 | 9 < SC ≤ 24 |

death | --- |

Disease State to From | S1 | S2 | S3 | S4 | Death |
---|---|---|---|---|---|

S1 | 2545 | 369 | 0 | 0 | 0 |

S2 | 879 | 3180 | 49 | 7 | 2 |

S3 | 1 | 83 | 99 | 18 | 1 |

S4 | 0 | 5 | 36 | 158 | 5 |

**Table 4.**Transitions of disease states observed during the first 15 days in the SepNet trials (MAXSEP/VISEP/SISPCT).

Disease State to From | S1 | S2 | S3 | S4 | Death |
---|---|---|---|---|---|

S1 | 172/109/425 | 28/28/35 | 3/3/6 | 0/0/1 | 1/0/1 |

S2 | 52/47/100 | 739/664/1308 | 62/88/155 | 2/4/5 | 1/2/3 |

S3 | 5/3/6 | 152/159/323 | 669/665/1286 | 60/85/142 | 2/3/17 |

S4 | 0/1/0 | 8/6/7 | 120/126/242 | 548/650/1245 | 24/24/49 |

Study | S1 | S2 | S3 | S4 |
---|---|---|---|---|

PROGRESS | 0.01 | 0.01 | 0.05 | 0.14 |

MAXSEP | 0.10 | 0.10 | 0.13 | 0.25 |

VISEP | 0.11 | 0.13 | 0.17 | 0.27 |

SISPCT | 0.11 | 0.15 | 0.20 | 0.30 |

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## Share and Cite

**MDPI and ACS Style**

Przybilla, J.; Ahnert, P.; Bogatsch, H.; Bloos, F.; Brunkhorst, F.M.; SepNet Critical Care Trials Group; PROGRESS study group; Bauer, M.; Loeffler, M.; Witzenrath, M.;
et al. Markov State Modelling of Disease Courses and Mortality Risks of Patients with Community-Acquired Pneumonia. *J. Clin. Med.* **2020**, *9*, 393.
https://doi.org/10.3390/jcm9020393

**AMA Style**

Przybilla J, Ahnert P, Bogatsch H, Bloos F, Brunkhorst FM, SepNet Critical Care Trials Group, PROGRESS study group, Bauer M, Loeffler M, Witzenrath M,
et al. Markov State Modelling of Disease Courses and Mortality Risks of Patients with Community-Acquired Pneumonia. *Journal of Clinical Medicine*. 2020; 9(2):393.
https://doi.org/10.3390/jcm9020393

**Chicago/Turabian Style**

Przybilla, Jens, Peter Ahnert, Holger Bogatsch, Frank Bloos, Frank M. Brunkhorst, SepNet Critical Care Trials Group, PROGRESS study group, Michael Bauer, Markus Loeffler, Martin Witzenrath,
and et al. 2020. "Markov State Modelling of Disease Courses and Mortality Risks of Patients with Community-Acquired Pneumonia" *Journal of Clinical Medicine* 9, no. 2: 393.
https://doi.org/10.3390/jcm9020393