Logical analysis of survival data: prognostic survival models by detecting high-degree interactions in right-censored data
1G-SCOP, Grenoble-Science Conception Organization and Production, 46, Avenue Viallet 38031, Grenoble, France and 2RUTCOR, Rutgers Center for Operations Research, 640 Bartholomew Rd., Piscataway, NJ 08854, USA
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 DESCRIPTION OF THE...3 METHODS: LASD4 EMPIRICAL RESULTS5 DISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Motivation: Survival analysis involves predicting the time to event for patients in a dataset, based on a set of recorded attributes. In this study we focus on right-censored survival problems. Detecting high-degree interactions for the estimation of survival probability is a challenging problem in survival analysis from the statistical perspective.
Results: We propose a new methodology, Logical Analysis of Survival Data (LASD), to identify interactions between variables (survival patterns) without any prior hypotheses. Using these set of patterns, we predict survival distributions for each observation. To evaluate LASD we select two publicly available datasets: a lung adenocarcinoma dataset (gene-expression profiles) and the other a breast cancer dataset (clinical profiles). The performance of LASD when compared with survival decision trees improves the cross-validation accuracy by 18% for the gene-expression dataset, and by 2% for the clinical dataset.
Availability: Executable codes will be provided upon request.
Contact: louis-philippe.kronek{at}g-scop.fr; areddy{at}rutcor.rutgers.edu
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 DESCRIPTION OF THE...3 METHODS: LASD4 EMPIRICAL RESULTS5 DISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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Survival analysis differs from regression mainly because of the presence of censored data. A sample is considered to be censored if it has incomplete time to event information. There are different types of censoring, the most common one being right-censoring. In this article we will concentrate on right-censored survival analysis problems. Right-censoring occurs when the sample did not have the event until the end of the study, their censoring time serves as a lower bound for their time to event. If the study was conducted for long enough periods then the event of interest would be observed in all patients.
Cox proportional hazards regression (Klein and Moeschberger, 2003) is a classical method for modeling the effects of covariates on the survival time. More recent techniques are those of decision trees for estimating survival functions (LeBlanc and Crowley, 1992), meta-classifiers such as bagging with survival decision trees as base classifiers is shown in several papers (Hothorn et al., 2004; Ishwaran et al., 2004; Ruczinski et al., 2003) an adpatation of Random >Forest for survival analysis is implemented by Breiman and Cutler (available at http://www.stat.berkeley.edu/~breiman/RandomForests/surv_manual.pdf).
A large number of publications consider different performance measures to evaluate the goodness of fit of the estimated survival function. The most standard one is concordance accuracy or c-statistic (Harrell et al., 1983). The c-statistic is equivalent to the area under the receiver operating characteristic (ROC) curve (AUC) when we disregard the censored samples. There are several publications (Ambler et al., 2002; Graf et al., 1999) which show that the c-statistic is a biased measure, and propose new performance measures such as the Brier score (Graf et al., 1999), the Sep and D measures (Royston and Sauerbrei, 2004), or loss functions (Hothorn et al., 2006).
In this study we present combinatorial tools for survival analysis problems. In particular, we have developed a new methodology to estimate right-censored survival functions which is based on the principles of Logical Analysis of Data (LAD). LAD is a data-mining technique which is a rule-based classification method. The main concepts of LAD have been introduced in Crama et al. (1988), and then were developed further in Alexe and Hammer (2006a,b), Bonates and Hammer (2006) and Goldberg and Shan (2007). It has been applied very successfully to a wide array of problems in the fields of medicine, finance, social studies, etc. (Alexe et al., 2002, 2004; Hammer and Bonates, 2006; Hammer et al., 1999).
There are a lot of data-mining approaches with the same background as LAD. One can mention multifactor dimensionality reduction (MDR) which has been successfully applied to detect gene–gene or gene–environment interactions (Hahn et al., 2003).
In Section 2, we present the problem description and the notations. We also explain the performance measure—Brier score—which we will use to evaluate the survival models. In Section 3, a novel rule-based methodology called Logical Analysis of Survival Data (LASD) is introduced. We present the construction of logical survival patterns and the estimation of the survival probability distribution for each observation. In Section 4, we use two publicly available benchmark datasets to illustrate the strength of the newly developed survival tools.
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2 DESCRIPTION OF THE SURVIVAL PROBLEM AND NOTATIONS |
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TOPABSTRACT1 INTRODUCTION2 DESCRIPTION OF THE...3 METHODS: LASD4 EMPIRICAL RESULTS5 DISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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The random variables
and 
represent the time to event and censoring time, respectively. Observed variables are represented by a triplet: (T,
,Z) where T=(,) and the censoring status =1 if the event occurred, and is 0 otherwise. Z represents the set of attributes. T is either exactly the time to event or a lower bound depending of the censoring status. The dataset consists of a sample of N observations from the above observed variables denoted by(ti,
i,zi)i=1...N.
A basic quantity in this kind of problem is the survival function S(t)=P(>t) i.e. the probability of surviving at least until time t. Let us denote by 
(t) an estimated survival function. One of the standard estimators for the survival function was given by Kaplan and Meier [1958] (Klein and Moeschberger, 2003). Let dt be the number of observations which experienced an event at time t, and let Yt be the number of observations which are at risk at time t (event at time t or later), then the formula for the Kaplan–Meier (KM) estimator is given as:
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| (1) |
In this article, we develop models to derive an estimator (t|zi of the survival function for each observation.
We will evaluate the accuracy of our estimation of survival function based on the Brier score. The Brier score was developed for predicting the inaccuracy of weather forecasting. This measure was later extended by Graf et al. (1999) to be used as a performance measure for survival problems incorporating censored samples. Brier score is a function of time. In the case when there are no censored observations in the data, for a given t, and observation i we expect that:
- If i had an event before time t, then we predict a low survival probability for i (close to 0).
- If i had an event after t, then we predict a high survival probability for i (close to 1).
In this case, for a given t, the Brier score is a measure of the squared deviation of survival estimate from the true probability. More precisely, in the absence of censored observations, the Brier score, BS(t), is defined as:
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Now let us consider the case when censoring is observed in the data. Let G(t)=P(>t) be the censoring distribution, the probability distribution of being censoring-free until time t. Let us denote by (
t) the estimate of G(t). In this case, the formula for Brier score is a weighted version of the above formula (without censoring), with the weights being 1/(ti) if an event occurs before time t, and 1/(t) if the event occurs after time t. Note that when the observation is censored before time t, it is not considered in the formula for the Brier score. We present below the formulation of the Brier score including the censored data:
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Integrated Brier Score (IBS) is an overall measure for the prediction of the model at all times.
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In our study, we use IBS as a performance measure to evaluate the survival distributions estimated by LASD.
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3 METHODS: LASD |
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TOPABSTRACT1 INTRODUCTION2 DESCRIPTION OF THE...3 METHODS: LASD4 EMPIRICAL RESULTS5 DISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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In this section we present a new methodology, LASD, which is a rule-based method, for estimating the survival distribution and event-risk of observations. This method is an extension of LAD for survival analysis problems.
LAD is a rule-based approach for 2-class classification problems. Using LAD, we can detect logical patterns, i.e. interactions between variables without a priori hypotheses (Boros et al., 2000). By combining the principles of LAD with optimization techniques one can build an accurate, small sized model in terms of the number of patterns and attributes. This is of great interest to understand the problem better, and to compare new conclusions with previous knowledge.
LASD is based on Boolean logic and is designed to handle binary attributes, just like LAD. The problem of transforming numerical attributes to binary ones, i.e. discretization, is well studied, and there exists many powerful methods which are surveyed in Liu et al. (2004) and Kotsiantis and Kanellopoulos (2006). It should be noted that discretization can result in loss of information. For attributes which take values in a finite set, like categorical attributes, we introduce a sufficient number of binary attributes to separate all the possible categories. The encoding scheme can be tuned for each categorical attributes. By default, we introduce one binary attribute per possible category. Smaller encoding schemes are possible, for example Ruczinski et al. (2003) proposed to use two binary attributes to represent single nucleotide polymorphisms (SNP) data which can take three possible values. A generalization of this encoding scheme involves using 
log2(n)
binary variables to encode a categorical attribute with n categories.
In the following, we introduce the concept of logical survival patterns and describe a procedure to generate them. Then we present an estimation of survival function for observations based on pattern coverage. Finally, we develop a method to build a survival model from the set of survival patterns.
3.1 Logical survival patterns
3.1.1 Definition
For each observation Oi, we consider that zi
{0,1}n and we denote by zi,k the value of the kth attribute of Oi. A logical survival pattern is defined as a subspace of {0,1}n described by a set of conditions on some attributes and containing observations with homogeneous survival properties. In other words, this set of conditions characterizes observations with similar time to event. We define the degree, deg(P), of a pattern P as the number of conditions necessary to characterize P. We denote by c1,c2,...,cdeg(P) the set of conditions defining P. A condition is a constraint on one attribute (e.g. zi,3=0). An observation Oi is said to be covered by pattern P if zi,j, (j=1,...,n), satisfies the following Boolean function:
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We denote by Cov(P) the set of observations covered by P and by P\c the pattern based on P without considering the condition c.
Now we introduce two specific kinds of survival patterns. We define a t-event pattern (EPt) as a pattern covering only observations that experience the event before t, i.e. ti
t and i=1. In a similar way, a t-event-free pattern (FPt) is defined as a pattern covering only those observations that experience an event after t, i.e. ti>t. Note that, EPt covers only observations that experienced an event, while FPt covers both events and censored observations.
In most practical problems, especially in biology and medicine, this definition of survival patterns is strict and unrealistic, and we relax the definition to: EPt is a pattern covering a major proportion of observations that experience an event before t, and a very small proportion of observations that experience an event after t. A symmetric relaxation applies for FPt. These proportions will be parameters of our pattern generation algorithm. Empirically, these parameters are tuned to maximize the performance in cross-validation experiments.
3.1.2 Pattern generation
The main idea of pattern generation is to build both EPt and FPt for the entire range of survival time. We use the idea of maximum pattern generation (Bonates and Hammer, 2006; Bonates et al., 2008) which is defined for a binary classification problem. This algorithm considers a reference observation and builds a pure pattern (covering only observations of the same class) that contains it and with the largest coverage. Generally, the constraint of pure pattern is relaxed to allow a small proportion of observations of the opposite class. We apply this idea in the survival context for a reference observation by looking for maximum EPt and FPt which covers it with the largest coverage.
Our pattern generation consists of generating these maximum survival patterns for each observation with an event in the dataset. Such a procedure is computationally very expensive. Our algorithm is based on an efficient heuristic, which is also described for classification problems in the same paper (Bonates and Hammer, 2006; Bonates et al., 2008). We will describe shortly the procedure to build one pattern. Then, we will introduce some useful notations to fully describe the procedure.
We consider a reference observation that experienced an event (i=1) at time ti. We illustrate the procedure for generating an EPti pattern for this observation. A pattern is built by backward selection. We start with a pattern that consists of conditions on all attributes which satisfy the reference observation. With a greedy procedure we remove conditions from the pattern description based on maximizing the separability power of the resulting pattern. The separability power ensures that the pattern is getting closer to observations we want to cover and farther from the others. At the end of each iteration, the number of observations covered increases. We keep removing conditions from the pattern description until the number of wrongly covered observation exceeds a user-defined parameter that we call the fuzziness. In the case of an EPti, the wrongly covered observations are observations with a time to event greater than ti. If you consider generating an FPti then the wrongly covered observations are observations which experienced an event (i=1) before time ti.
We introduce the following definitions: observations which had an event before time t will be denoted by Ot={Oi|tit and i=1} and O<t={Oi|ti<t and i=1}. In a symmetric way, we represent observations which had an event after time t by O
t={Oi|tit} and O>t={Oi|ti>t}. So an ideal EPt covers only observations from the set Ot and not from O>t. One can note that observations with tit and i=0 are not in either set. For an EPt, we call the set Ot as positive observations and O>t as negative observations. The reverse remark holds for FPt. We will refer to EPt t-event patterns and FPt t-event-free patterns as t-patterns when it is not necessary to make a distinction between them.
Disagreement between observation Oi and t-pattern P, DOi,P, is defined as the number of conditions in the pattern P not satisfied by observation Oi.
Disagreement between set of observations S and t-pattern P, DS,P, is defined as the weighted sum of DOi,P where OiS. The weights are the absolute differences between ti andt.
Separability power of t-pattern Pt, Sep(Pt), is defined as the ratio of the disagreement between t-pattern and the negative observation set and disagreement of t-pattern with the positive observation set.
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t,P is small. This denotes that the disagreement with observations that EPt can cover is low, and/or it is high with observations that are wrongly covered. Figure 1 describes the full heuristic to generate an EPt for a given reference observation (with time to event t), dataset and fuzziness. The algorithm for generating an event-free survival pattern (FPt) is symmetric to the one described above. In the next section, we will use these t-event patterns to estimate the survival distribution of observations.
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With this pattern generation method, we generate a large set of patterns (twice the number of observations with an event). In order to reduce redundant information and the effects of over-fitting, we want to identify from this large set a small subset of patterns that has a high performance. This filtering method is discussed in detail in Section 3.3. In the next section, we discuss how to estimate the survival function for an observation based on a set of selected survival patterns.
3.2 Survival function estimator
Based on the set of selected survival patterns, in this section we estimate the survival function of an observation. It is important to note here that an observation can be covered by an arbitrary number of patterns. We introduce the following concepts:
The estimated baseline survival function, b(t), is estimated as the KM estimate of all the observations in training set.
The estimated pattern survival function, p(t), is computed as KM estimate of Cov(P), set of observations covered by pattern P.
The survival function of an observation, (t|zi) covered by patterns P1,...,Pp is estimated as:
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For this estimation, B(t) is used as baseline. Then this baseline is corrected for observations based on pattern coverage. For an observation, we predict the survival function to be the mean of the survival estimates of all patterns covering it for each time point including the baseline survival. The baseline estimate is the default estimate for an observation not covered by any pattern.
To illustrate the above explanation, we present an example of an observation which is covered by two patterns in Figure 2. The KM survival functions of patterns 1, 2 and the baseline are shown in the figure. The survival function of that observation is estimated as the average of the survival probabilities of the two patterns and the baseline at all t (average of the three curves).
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Apart from the survival function, it is also possible to provide a single measure to evaluate the risk of an observation, as the area under the predicted survival curve.
3.3 Survival model
We define a survival model as a minimal subset of patterns which covers all observations with an event. The motivation is to reduce redundant information and to find a simple model to allow practical interpretation by experts in the field of the application.
We use the Brier score as the objective function for finding such a subset. Such a procedure is computationally very expensive. Instead of solving it to optimality, we use a greedy heuristic to find such a model. This heuristic is based on forward selection. We start with an empty set of patterns, and add patterns one by one to minimize the IBS. This procedure terminates either (i) when we have covered all observations with an event in the dataset at least a fixed number of times ( 1) or (ii) the amount of increase in the objective function, when we select another pattern to be added to the model, is smaller than some user-defined stopping parameter. The smaller the increase in the objective function, the lower is the new pattern's contribution to the model. This also addresses the problem of over-fitting, which is a frequent problem in biomedical datasets. Also, in the case of using the latter condition as the stopping criterion, it may occur that some of the observations are not covered by any pattern in the model. These observations are estimated to have the baseline survival curve.
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4 EMPIRICAL RESULTS |
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TOPABSTRACT1 INTRODUCTION2 DESCRIPTION OF THE...3 METHODS: LASD4 EMPIRICAL RESULTS5 DISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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In this section we will present the results of analyzing two publicly available benchmark datasets using LASD. The datasets were discretized and binarized by transforming each numerical variable into five binary variables by using unsupervised equal frequency cut-points (1/6th of the possible values lie between two consecutive cut-points). The performance of the survival models will be evaluated using IBS, as discussed in Section 2. Finally we compare the results of our new models with that of KM estimates and survival trees. We use the bagging function in the R-package ipred (Peters and Hothorn, 2007) for running the experiments for the survival trees with the parameter nbagg=1 (a single survival tree). Other parameters of the survival trees model were extensively tuned for minimizing IBS. We will cross-validate our method by running five random 5-folding experiments. We use the same five 5-folds for running KM estimates and survival trees.
4.1 Lung adenocarcinoma dataset (LAC)
This dataset was first analyzed by Beer et al. 2002). The dataset consists of the gene-expression profiles of 86 lung cancer patients based on microarray analysis. The authors identified a subset of 50 genes which showed a strong linkage to survival based on the gene-expression profiles. We have further analyzed these 50 genes using LASD.
4.2 German Breast Cancer Study Group 2 (GBSG2)
The GBSG2 dataset was first studied by Sauerbrei et al. (1999). This dataset contains 686 observations described by eight variables. The variables—age, menopausal status, tumour size, tumour grade, number of positive lymph nodes, hormonal therapy, progesterone and estrogen receptor concentrations—were investigated in 686 patients, of whom 299 had an event for recurrence-free survival and 171 died. This dataset was also used to test other methods: survival trees and survival ensembles (Hothorn et al., 2004, 2006; Sauerbrei et al., 1999).
4.3 Computational results
The average IBS for cross-validation experiments (five random 5-folding) using KM estimates, survival trees and LASD are shown in Table 1.
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It also contains the 95% confidence intervals. For LAC dataset it can be seen that LASD provides a huge improvement over KM estimates and survival trees. LASD improves the IBS by 26.9% when compared to the KM estimate, and 18.1% with survival trees. For GBSG2, the improvements are 10.6% and 1.7% for KM estimates and survival tree, respectively. Moreover, LASD models for both datasets have the smallest confidence interval when compared to the KM estimator and survival trees. Thus, compared to survival trees and KM estimates, LASD results are superior and more robust.
The description of the patterns in the LASD model for the LAC and GSGB2 dataset are shown, respectively, in Table 2 and Table 3. Each model consists of six patterns: four EPt and two FPt patterns for LAC, a three EPt and three FPt patterns for GSGB2. The survival patterns are described as simple Boolean formulae, for instance: EP2 in Table 3 is described as: positive nodes >4 AND progesterone receptor 50. The expected survival time and log-rank test statistic for the patterns are also presented in the tables. We see that the p-values associated with the log-rank test on the patterns are very low (<0.005). This is a good indication that the survival patterns have significantly distinct survival characteristics.
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Another indication of quality of the survival patterns is the plot of KM estimates built with the set of observations covered by each of the patterns (Figs 3 and 4). The plot of the survival function for the whole data is shown as reference (baseline). On each plot, one can observe the difference between EPt and FPt patterns in terms of survival properties. It can be seen that the curves of EPt are much lower than the baseline curve, showing that they cover high-risk observation for the event. The symmetric remark can be made for FPt which always have a survival probability above the reference baseline curve. These patterns characterize low-risk populations.
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5 DISCUSSION AND CONCLUSIONS |
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TOPABSTRACT1 INTRODUCTION2 DESCRIPTION OF THE...3 METHODS: LASD4 EMPIRICAL RESULTS5 DISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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LASD is an accurate prognostic tool for the estimation of survival functions. From the empirical studies, it is shown that the IBS for the LASD model are superior to those of survival trees and KM estimates. Also, the confidence intervals of the resulting predictions are very small, indicating the robustness of the resulting LASD model.
LASD algorithm is very easy to tune. It has only one parameter (fuzziness) for generating survival patterns, and an additional parameter (stopping criterion) for building the model.
The main advantage of LASD is that compared to classic statistical tools it can detect interactions between attributes (patterns), without any prior hypotheses. Survival patterns are meaningful characterizations of groups of observations which are homogenous in terms of survival. The p-values associated with the log-rank tests for the group of observations covered by a pattern and the rest of the data are very low (<0.005), supporting the above statements. Survival patterns EPt characterize a high-risk population, while FPt characterize a low-risk population.
Survival patterns can be represented by simple rules. They are transparent objects, and can be easily understood by doctors and biologists. They are very useful biological research hypotheses and can be interpreted and further investigated by the experts. For instance, in the case of gene-expression profiles, patterns can detect novel interactions of genes which are linked with survival.
Each survival pattern explains only a smaller proportion of the dataset. In order to explain the entire dataset, we need a survival model, i.e. a group of survival patterns. Patterns can be combined in an infinite number of ways to build a ‘good’ model. In this article, we present a general framework where the emphasis is on building concise models containing both EPt and FPt patterns. Each pattern in the model has the same weight, in this framework, for estimating the survival distribution. This general framework can be modified based on the particular application. An interesting problem for future work is to device more sophisticated weighting schemes for patterns in the estimation of survival distributions.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 DESCRIPTION OF THE...3 METHODS: LASD4 EMPIRICAL RESULTS5 DISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSREFERENCES |
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We are grateful to our advisors Gabriela Alexe, Nadia Brauner and Endre Boros for their constant support, guidance and patience during the course of this study. The first author gratefully acknowledges the partial support provided by the program Exploradoc of Region Rhone Alpes (France).
Conflict of Interest: none declared.
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