Use of a multistate model in survival predictions in cardiology studies

INTRODUCTION

In clinical research, statistical methodologies used for survival analyses range from the easiest non-parametric models—such as Kaplan-Meier estimates—to semi-parametric models such as the Cox proportional hazards model.1 When multiple adverse events are of interest, it is common practice to create a composite variable, such as major adverse cardiovascular events (MACE), to indicate whether a patient experienced any event at the follow-up, facilitating the application of these models,2 which provide numerous advantages with some associated limitations; for instance, they usually only consider the time to the index event for each patient, regardless of which component of the composite variable triggered such an event, which complicates the interpretation of the intervention effect since reliable conclusions cannot be drawn about the effects on individual components due to potential bias from competing risks.3,4

For this reason, multi-state models have gained traction in recent years, providing a framework to analyze disease progression.4 In medical applications, states in a multi-state model can represent various adverse events that patients may experience over time.5 A multi-state model is defined by its state structure, consisting of states and transitions across them. This structure allows for defining certain states as absorbing (from which the patient cannot exit, such as death) or transient (intermediate states between the initial and the absorbing states). These models extend competitive risk models—a multi-state model with 1 initial and multiple mutually exclusive absorbing states—by enabling any state structure, for example, extending analysis to what happens after a transient event.6 Additionally, they allow assessment of variables that impact the patients’ probability of transitioning from one state to the other by modeling these transitions, which is particularly useful in long-term clinical trials.7

Multi-state models can incorporate several covariates, such as demographic characteristics or biomarkers, to assess their effects on event rates and time-to-event. This aids in identifying risk factors and understanding their impact on patient prognosis, thus facilitating the efficacy evaluation of various treatments or interventions, and the selection of the most suitable strategy for each patient.8-10

The aim of this study is to model disease progression in the patients of a cardiology study by using a multi-state model and evaluating its applicability and limitations.

METHODS

Data

The database used is the updated version of the multicenter and retrospective PACO-PCI registry (Antithrombotic strategies in elderly patients with atrial fibrillation revascularized with drug-eluting stents),11 which included a total of 1057 patients older than 75 years with atrial fibrillation on oral anticoagulant therapy after revascularization with drug-eluting stents from 2015 through 2019. The endpoints of this registry included MACE (death, acute myocardial infarction [AMI], revascularization) and bleeding 12 months after treatment. Updated data extend patient follow-up to 5 years. Previous results from the PACO-PCI study11 demonstrated the efficacy of various antithrombotic therapies regarding the onset of MACE and major bleeding events. This study uses such data to illustrate the application of multi-state models, focusing on factors influencing the occurrence of the events of interest to achieve a more individualized model of disease progression.

Data analysis

The multi-state model used includes 4 states (1 initial state, 2 transient ones, and 1 absorbing state) and the possible transitions across them (figure 1). Specifically, a patient enters initial state 1 (treatment) at the time of the intervention. From this state, they can transition to transient state 2 (bleeding) if a major bleeding event occurs, transient state 3 (AMI/revascularization) if they experience an AMI or require re-intervention, or absorbing state 4 (death) if they die. From state 2 (bleeding), patients can transition to state 3 (AMI /revascularization) if they experience an AMI or require re-intervention or vice versa if a new bleeding event occurs. Patients can transition to the absorbing state from any state if they die. Compared with traditional methods—event composition and competing risks—this model distinguishes the severity of adverse events while maintaining a certain simplicity.

Figure 1. Proposed multi-state model. Upon intervention, patients enter the initial state 1 (treatment), from which they can move to transient state 2 (bleeding) if they experience a major bleed, or transient state 3 (acute myocardial infarction [AMI] or revascularization) if they experience an AMI or require re-intervention, or state 4 (absorbing, death) if they die. Patients in state 2 (bleeding) can transition to state 3 (AMI or revascularization) if they experience an AMI or require re-intervention. Patients in state 3 (AMI or revascularization) can move to state 2 (bleeding) if they experience a new bleed and can also transition to state 4 (death) if they die. The number of patients experiencing each adverse event is indicated alongside each transition, based on an initial cohort of 1057 patients. CKD, chronic kidney disease; LMCA, left main coronary artery; LVEF, left ventricular ejection fraction.

The multi-state model was adjusted using the msm12 package for R,13 which employs an exponential model for the time spent in each state. This package allows fitting a general multi-state model to survival data, requiring a complete data matrix; missing data for quantitative variables were completed with the corresponding mean value.

The proposed model for the time spent in each state allows including factors affecting each transition. For complete model determination purposes, variables associated with each transition were selected, ie, factors influencing the probability of transitioning from one state to the other. We chose a starting set of variables that could impact the occurrence of adverse event based on clinical criteria, as shown in table 1 of the supplementary data. This set includes the most important baseline characteristics, the number of vessels treated, and the scores obtained on the PreciseDAPT,14 HAS-BLED,15 and CHA²DS²-VASc16 scales. Afterwards, a multi-state model was adjusted including these variables each one at a time to identify their individual influence on each transition. Results of this analysis are shown in table 1 of the supplementary data. Subsequently, we tested different combinations of influential variables to achieve models with the best fit based on the Akaike information criterion, which favors model fit with the fewest covariates.12 Finally, we selected the model with the lowest value for this criterion that provided the most clinically relevant information.

Table 1. Covariates selected for the survival model in each state transition

Afterwards, we conducted a goodness-of-fit study of the multi-state model to determine whether the exponential model adequately fit the observed time in each state. This analysis revealed that the model overestimates event-free survival after 1000 days (slightly more than 2.5 years), mainly because it underestimates the prevalence of death beyond this period. Therefore, a maximum follow-up of 1000 days was considered for the final analysis.

Furthermore, we conducted a traditional survival analysis to compare it with our model. Specifically, a Cox regression model was fitted for the MACE variable, defined as AMI, revascularization, bleeding, or death. We conducted Univariate Cox regression analyses to determine factors affecting the occurrence of MACE (table 1 of the supplementary data) using the same initial set considered for the multi-state model. Based on these results, we selected a subset of variables to adjust a multiple Cox regression model. Again, the Akaike information criterion was used to select the best model among all possible models. We performed all calculations with the statistical program R, version 4.1.1; in particular, the Survival17 package was used for the above-mentioned traditional survival analysis.

RESULTS

The database includes information on 20 Spanish centers for a total of 1057 patients older than 75 years who underwent percutaneous coronary intervention with drug-eluting stents from 2015 through 2019. The patients’ mean age is 81 ± 4.2 years, and most (almost 70%) are men. Diabetes mellitus—a known risk factor for various cardiovascular diseases—was present in 42.4% of the population, and most patients (about 80%) had experienced a prior cardiovascular event. The patients’ baseline characteristics are shown in table 2 of the supplementary data. Only 5 variables had missing data: anemia (4.4%), chronic kidney disease (CKD) (0.9%), left ventricular ejection fraction (LVEF) (3.2%), PreciseDAPT score (0.6%), and treated left main coronary artery disease (LMCAD) (< 0.1%).

Table 2. Results of the Cox multiple regression model for major cardiovascular adverse events (1000-day follow-up)

The mean follow-up was 854.8 days (2 years and 4 months), with the shortest follow-up being 2 days and the longest one, 2018 days. Figure 1 and table 3 of the supplementary data illustrate that death is the most common event among patients (14.1%), followed by major bleeding (10.6%). After the intervention and stent treatment, a significant number of patients experience a new AMI or require re-intervention (7.9%) as their first adverse event.

Table 3. Characteristics (risk factor values) of hypothetical patients used to demonstrate the predictive capabilities of the multistate model

Multi-state model

We obtained an estimate and 95% confidence interval (95%CI) of the hazard ratio (HR) for each variable and transition following the variable selection process. Table 1 shows the estimated risk associated with each variable in each transition. For the PACO-PCI study data, the resulting multi-state model revealed, for example, that a higher score on the PreciseDAPT scale increases the risk of bleeding after treatment (HR, 1.05; 95%CI, 1.03–1.06), and that LVEF is a protective factor vs after bleeding (HR, 0.95; 95%CI, 0.92–0.97). The transition from treatment to death is influenced by the number of vessels treated and LVEF, and by the PreciseDAPT and HAS-BLED scores. The transition from treatment to bleeding is related to anemia and the PreciseDAPT score. After a bleeding event, the likelihood of experiencing a new AMI or revascularization is associated with treated LMCAD. The transition from bleeding to death depends on the LVEF and previous coronary artery bypass graft. The transition from treatment to AMI or revascularization is related to diabetes and the PreciseDAPT score. After a new AMI has occurred or revascularization has been performed, the likelihood of bleeding is influenced by the HAS-BLED score. Lastly, the transition from AMI or revascularization to death is determined by CKD and LVEF.

Comparison between the multi-state model and Cox regression analysis

The results of the MACE variable study with a Cox regression analysis—which provides the HR for each MACE predictor—are shown in table 2. The factors included in the best multiple Cox regression model were diabetes, CKD, anemia, the PreciseDAPT and HAS-BLED scales, LVEF, and the number of vessels treated (Table 2).

Although the variables treated LMCAD and previous coronary artery bypass graft were not significant predictors of MACE in the multiple Cox regression analysis, they were significant for some transitions in the multi-state model. Diabetes, CKD, anemia, the PreciseDAPT and HAS-BLED scales, LVEF, and the number of vessels treated were significant predictors in the univariate Cox regression analysis (table 1 of the supplementary data) and for some transitions in the multi-state model.

Utility of the multi-state model

In contrast to the Cox regression model, a fitted multi-state model, like the one proposed, can predict the probability of a patient transitioning across states within a specified period of time, that is, the probability of experiencing each type of event after treatment or after experiencing another transient event within a specified timeframe. For example, it is possible to calculate the probability that a patient with certain baseline characteristics who has experienced major bleeding will die within 1 year.

To illustrate the predictive capability of the model, we defined 2 types of patients—low- and high-risk—whose characteristics are shown in table 3. We used the multi-state model to predict the probability of each of these hypothetical patients in each of the possible transitions within the first year after treatment or after an exit event. These predictions for the 2 types of patients are shown in figure 2. For example, the PACO-PCI data reveal that that the probability rates of death 1 year after major bleeding are 75% and 10% for high- and low-risk patients, respectively.

Figure 2. Event-free survival graphs (A, E) and survival graphs (B, C, D, F, G) show the probability of low- (green) and high-risk (red) patients experiencing an adverse event within 1000 days (a little more than 2.5 years). A: probability of bleeding after treatment. B: probability of experiencing a new acute myocardial infarction or revascularization after treatment. C: probability of death after treatment. D: probability of a new acute myocardial infarction or revascularization after bleeding. E: probability of bleeding after a new acute myocardial infarction or revascularization. F: probability of death after bleeding. G: probability of death after a new acute myocardial infarction or revascularization.

DISCUSSION

Results demonstrate the added value of multi-state models in survival analyses within biomedical research. Multi-state models introduce additional predictive variables beyond those identified by traditional survival analyses, and provide information on the expected time and probability of transitioning from one state to the other based on risk factors, treatment characteristics, and previous disease progression. Traditional analyses only provide information on general significant variables, without clarifying which specific adverse event they predict.

In a prior study, a multi-state model with a 3-state structure was applied to data from the Synergy ACS study,9 selecting the most determinant variables for each type of adverse event. Specifically, diabetes mellitus, the number of diseased vessels, and CKD were analyzed in relation to the time elapsed from treatment administration to the occurrence of a new AMI or revascularization; age, LVEF, and previous percutaneous coronary intervention for the time elapsed from treatment administration to death; and diabetes mellitus, the number of diseased vessels, and stent thrombosis for survival from post-treatment AMI or revascularization. In the PACO-PCI study data10 given the patients’ advanced age and baseline characteristics, we observed a high probability of bleeding after treatment, so this variable was included as a transient state in the model. There are common predictors in the 2 studies, such as the number of diseased or treated vessels, LVEF, and CKD, though not all factors corresponded to the same transition in the multi-state model. Moreover, it is notable that each database includes unique variables not found in the other.

In the current dataset, factors such as age and stent thrombosis are not statistically significant due to the patient profile and the fact that most experienced a prior adverse event. Consequently, predictive scales such as the Precise-DAPT and the HAS-BLED—which include multiple events—are crucial regarding adjusting the model.

Multi-state models have been used in other cardiology studies2,6,18-20 with different state structures. In heart failure, using the model applied by Upshaw et al.18, both LVEF and diabetes mellitus were found to be predictors of death. CKD is related directly to death and to death following hospitalization for heart failure. Postmus et al.1 used a multi-state model that was similar to the disability model to predict hospitalization for heart failure and death, identifying AMI, diabetes mellitus, LVEF, and CKD as predictors.

The proposed multi-state model has certain limitations. Regarding data, it is a retrospective observational registry affected by the limitations of all observational studies. Specifically, the most significant limitations of this study are: 1) the heterogeneity of follow-up, which can introduce significant biases; 2) its limited statistical power for a model with 7 transitions; and 3) its retrospective design without event adjudication, implying that many deaths may have been due to unreported ischemic or hemorrhagic events. It is also worth noting that the variables included in the model were selected not only based on statistical criteria but also subjectively by the researcher, meaning that results should be interpreted with caution. Although the management of missing data through multiple imputation would have accounted for variability due to data loss, model selection with missing data in multi-state models has not yet been resolved in the literature.

Finally, multi-state models are currently widely used in fields outside the cardiovascular clinical trial,2 hematology,21 and oncology settings.22,23 Despite their proven utility, there are 3 main limitations in performing multi-state model analysis. First, the msm package in R assumes the Markov property, meaning that in our model, survival after a transient event does not depend on the time from the initial intervention to the corresponding event. Second, multi-state models require sufficient observed events to have statistical power and make reliable predictions. Third, most software for multi-state model analysis is integrated into statistical packages and is not easy to use; for example, each requires a different data structure. Interested readers can consult a systematic review of existing programs.24

CONCLUSIONS

Multi-state models are essential for describing disease progression due to their capacity to adapt to various events or factors through their state structure. Another advantage is that they consider all available follow-up data, including patients who may have or experienced an event. Additionally, they provide information on the estimated time to an event along with the probability of transitioning across states, making them an essential tool in cardiovascular event analysis by providing more accurate estimates of future event risk.

FUNDING

None declared.

ETHICAL CONSIDERATIONS

The original study (PACO-PCI) was approved by the reference CEIm of the Health Areas of León and El Bierzo (Spain) on 11-26-2019, reference no. 19167. Since this study involved new statistical analysis of observed results without new tests or data collection, ethical committee review was deemed unnecessary.

STATEMENT ON THE USE OF ARTIFICIAL INTELLIGENCE

No artificial intelligence was used in the development of this article.

AUTHORS’ CONTRIBUTIONS

All authors contributed equally to the design of the multi-state model. J.M. de la Torre-Hernández and J.L. Ferreiro provided the data. N. Montoya and A. Quirós conducted the data analysis and model implementation. N. Montoya, A. Quirós, and A. Pérez de Prado drafted the manuscript, and all authors substantially contributed to the review process.

CONFLICTS OF INTEREST

J.M. de la Torre-Hernández is the editor-in-chief of REC: Interventional Cardiology; A. Pérez de Prado is an associate editor of REC: Interventional Cardiology; in both cases, the journal’s editorial procedure to ensure impartial handling of the manuscript has been followed. The remaining authors declared no conflicts of interest whatsoever.

ACKNOWLEDGMENTS

We wish to thank all researchers of the PACO-PCI registry.

SUPPLEMENTARY DATA

REFERENCES