A Deep Learning Approach for Dynamic Survival Analysis with Competing Risks

1 INTRODUCTION . Survival analysis informs our understanding of the relationships between the ( distribution of ) first hitting times of events ( such as death , onset of a certain disease , etc . ) and the covariates , and enables us to issue corresponding risk assessments for such events . Clinicians use survival analysis to make screening decisions or to prescribe treatments , while patients use the information about their clinical risks to adjust their lifestyles in order to mitigate such risks . However , designing the best clinical intervention for a patient is a daunting task , as the appropriate level of interventions or the corresponding outcome often depend on whether this patient is susceptible to or suffers from competing risks . For example , studies in ( Koene et al . ( 2016 ) ) have shown that various treatments , such as chemotherapy , for breast cancer increase the risk of a cardiovascular events . To refer to the same example in cystic fibrosis ( CF ) , the decision on lung transplantation , which is particularly recommended for patients with end-stage respiratory failure , must jointly account for deaths from other CF-associated failures ( e.g. , CF-associated liver failure ) since they share a number of risk factors ( Kobelska-Dubiel et al . ( 2014 ) ) . Meanwhile , as a growing number of electronic health records ( EHRs ) have been deployed in hospitals1 , modern clinical data in EHR often comprises longitudinal measurements ; especially with chronic diseases , patients are followed up over the span of years , usually as part of regular physical examinations . Information contained in these longitudinal ( follow-up ) measurements is of significant importance such that interpreting these measurements in their historical context can offer an explanation for how the underlying process of clinical events progresses ( Rizopoulos et al . ( 2017 ) ; Suresh et al . ( 2017 ) ) . For example , forced expiratory volume ( FEV1 ) , and its development , is a crucial biomarker in assessing the severity of CF as it allows clinicians to describe the progression of the disease and to anticipate the occurrence of respiratory failures ( Nkam et al . ( 2017 ) ; Li et al . ( 2017 ) ) . Therefore , to provide a better understanding of disease progression , it is essential to incorporate longitudinal measurements of biomarkers and risk factors into a model . Rather than 1EHRs are deployed in more than 75 % of hospitals in the United States , according to the recent data brief by the Offcie of National Coordinator ( ONC ) ( Henry et al . ( 2016 ) ) . discarding valuable information recorded over time , this allows us to make better risk assessments on the clinical events of interest . In light of the discussion above , we design a deep neural network for dynamic survival analysis with competing risks , which we call Dynamic-DeepHit , that learns , on the basis of the available longitudinal measurements , a flexible data-driven distribution of first hitting times of various events of interest . An important aspect of our method is that it naturally handles situations in which there are multiple competing risks where more than one type of event plays a role in the survival setting . To enable survival analysis with competing risks , Dynamic-DeepHit employs a network architecture that consists of a single shared subnetwork and a family of cause-specific subnetworks . We incorporate historical information of each patient using a recurrent neural network ( RNN ) structure in the shared subnetwork , which allows our method to update personalized and cause-specific risk predictions as additional measurements are collected in a fully dynamic fashion . We conduct a set of experiments on real-world datasets showing that our model outperforms state-of-the-art survival models in discriminating individual risks for different causes . In addition , we provide variable importance and dynamic risk predictions of our method to yield clinical usefulness in interpreting the associations between covariates and the survival . Related works : Since the Cox proportional hazard model ( Cox ( 1972 ) ) was first introduced , a variety of methods have been developed for survival analysis , ranging from statistical models to deep learning techniques ( Fine & Gray ( 1999 ) ; Ishwaran et al . ( 2008 ) ; Lee & Whitmore ( 2010 ) ; Yu et al . ( 2011 ) ; Lee et al . ( 2018 ) ; Luck et al . ( 2017 ) ; Katzman et al . ( 2016 ) ; Alaa & van der Schaar ( 2017 ) ; Bellot & van der Schaar ( 2018 ) ) . Especially , deep networks have been shown to achieve significantly improved performance in survival analysis ( Katzman et al . ( 2016 ) ; Luck et al . ( 2017 ) ; Alaa & van der Schaar ( 2017 ) ; Bellot & van der Schaar ( 2018 ) ; Lee et al . ( 2018 ) ) owing to the ability to represent complicated associations between features and outcomes . However , all of these methods provide static survival analysis : they use only the current information to perform the survival predictions and most of the works focus on a single risk rather than multiple risks . In this paper , we extend the idea of ( Lee et al . ( 2018 ) ) , which finds the estimated joint distribution of the first hitting time and competing events utilizing a deep network , into the longitudinal setting where repeated measurements are available . Few methods have been developed to use the longitudinal time-to-event data in order to enable dynamic survival analysis under competing risks . The first strand of literatures includes landmarking ( Heagerty & Zheng ( 2005 ) ; Zheng & Heagerty ( 2005 ) ; van Houwlingen ( 2007 ) ; van Houwelingen & Putter ( 2008 ) ) and joint models ( Henderson et al . ( 2000 ) ; Ibrahim et al . ( 2010 ) ; Tsiatis & Davidian ( 2004 ) ; Brown et al . ( 2005 ) ; Barrett & Su ( 2017 ) ) . Landmarking refrains from modeling the time-dependent aspect of longitudinal variables and , instead , obtains survival probabilities from the survival model fitted to subjects who are still at risk at the time points of interest ( i.e. , landmarking times ) . On the other hand , joint models explicitly model the longitudinal process and leverage their predictions as inputs in a separate survival process used for predicting survival probabilities . However , both approaches , and their variations ( van Houwelingen & Putter ( 2008 ) ; Tsiatis & Davidian ( 2004 ) ; Brown et al . ( 2005 ) ; Barrett & Su ( 2017 ) ) , make strong assumptions about the underlying stochastic models for the survival process ( in landmarking ) or for both the longitudinal and survival processes ( in joint models ) . Hence , model mis-specifications ( e.g. , typically , a linear mixed model and a Cox proportional hazard model ) limit their ability to learn and infer complex interactions between covariates and survival times , which are common in many diseases with heterogeneity . The second strand models the longitudinal data using variants of RNNs ( Choi et al . ( 2016a ; b ) ; Razavian et al . ( 2016 ) ; Lipton et al . ( 2016 ) ) and avoids the need for explicit model specifications , which results in performance gain in terms of predictive accuracy . However , these models can not properly cope with time-to-event data where the goal is to find the probability of the first hitting event occurring at different times of our interest . Instead , given longitudinal measurements , they view making risk predictions of one or multiple event ( s ) as solving a single or multiple label ( s ) classification problem at each time stamp , e.g. , whether an event occurs or not at the measurement time ( Choi et al . ( 2016a ; b ) ) , within a predefined time-window ( Razavian et al . ( 2016 ) ) , or at the end of the available longitudinal measurements ( Lipton et al . ( 2016 ) ) . To our best knowledge , this paper is the first to investigate a deep learning approach for dynamic time-to-event analysis with competing risks on the basis of repeated measurements ( longitudinal time-to-event data ) . 2 BACKGROUND . 2.1 LONGITUDINAL TIME-TO-EVENT DATA . We consider a dataset D = { ( X i , τ i , ki ) } Ni=1 comprising time-to-event ( survival ) data for N subjects who have been followed up for a certain amount of time . For each subject i , X i is a set of longitudinal observations including both static and time-varying covariates , τ i = min ( T i , Ci ) is the time information with T i ∈ T indicating the time-to-event and Ci ∈ T indicating the time-tocensoring , and ki ∈ K being the event or censoring that occurred at τ i . Define X i ( t ) = { xi ( tij ) : 0 ≤ tij ≤ t for j = 1 , · · · , M i } a set of longitudinal observations until time t where xi ( tj ) is covariates recorded at time tj . Here , we distinguish notations between time stamps j = 1 , · · · , M i and the corresponding actual times tij = ti1 , · · · , tiMi , since measurements are not necessarily observed at regular intervals . Then , we use X i = X i ( tiMi ) to denote a whole set of longitudinal observations available for subject i until the last measurement time tiMi of that subject for notational simplicity . A set of possible survival times is denoted as T = { 0 , 1 , · · · , Tmax } with Tmax being a predefined maximum time horizon , where we treat survival time as discrete2 ( e.g. , a resolution of one month ) and the time horizon as finite ( e.g. , no patients lived longer than 100 years ) . A set of possible events is K = { ∅ , 1 , 2 , · · · , K } , with ∅ denoting right-censoring as survival data is frequently right-censored due to subjects being lost to follow-up . We assume that every subject experiences exactly one event among K ≥ 1 possible events of interest within T . This includes cause-specific deaths due to CF , where deaths from other causes are competing risks for death due to respiratory failure ( Gooley et al . ( 1999 ) ) . Figure 1 depicts a time-to-event dataset comprising histories of longitudinal measurements with competing risks , where subjects are aligned based on the synchronization event . The aforementioned characteristics are highlighted with annotations . 2.2 CUMULATIVE INCIDENCE FUNCTION . The cause-specific cumulative incidence function ( CIF ) is key to survival analysis under the presence of competing risks . As defined in ( Fine & Gray ( 1999 ) ) , the CIF expresses the probability that a particular event k∗ ∈ K occurs on or before time τ∗ conditioned on the history of longitudinal measurements X ∗ . The fact that longitudinal measurements have been recorded up to t∗M∗ implies survival of the subject up to this time point . Thus , the CIF is defined as follows : Fk∗ ( τ ∗|X ∗ ) , P ( T ≤ τ∗ , k = k∗|X ∗ , T > t∗M∗ ) = ∑ m≤τ∗ P ( T = m , k = k∗|X ∗ , T > t∗M∗ ) . ( 1 ) Whenever a new measurement is recorded for this subject at time t > t∗M∗ , we can update ( 1 ) accounting for that information in a dynamic fashion . 2Discretization is performed by transforming continuous-valued times into a set of contiguous time intervals , i.e. , T = τ implies T ∈ [ τ , τ + δt ) where δt implies the resolution . Similarly , the survival probability of a subject at time τ∗ given X ∗ ( i.e. , the probability that a subject does not experience any event before or at time τ∗ ) can be derived by S ( τ∗|X ∗ ) , P ( T > τ∗|X ∗ , T > t∗M∗ ) = 1− ∑ k 6=∅ Fk ( τ ∗|X ∗ ) . ( 2 ) Our goal is to estimate the CIF , F̂k∗ ( τ∗|X ∗ ) , from the dataset D , in order to analyze the causespecific risk given the history of observations and to issue dynamic risk predictions .

Summary: The authors propose Dynamic-DeepHit, a survival analysis framework for modeling longitudinal data with multiple competing risks. As opposed to previous works, Dynamic-DeepHit can model survival events (e.g. death, cancer relapse) which can be driven by multiple, potentially competing, underlying risks. The proposed model uses an RNN shared across multiple risks for processing past-to-recent measurements, and multiple feedforward nets that accept the most recent measurements and the hidden layer of shared RNN. Joint predictions (across time and competing risks) are made using a softmax layer. The model is tested on two datasets where Dynamic-DeepHit outperforms other baselines.