Modeling treatment events in disease progression

1 INTRODUCTION . The course of disease progression in individual patients is one of the biggest uncertainties in medical practice . In an ideal world , accurate , continuous assessment of a patient ’ s condition helps with prevention and treatment . However , many medical tests are either harmful or inconvenient to perform frequently , and practitioners have to infer the development of disease from sparse , noisy observations . In its simplest form , the problem of modeling disease progressions is to fit the curve of y ( t ) , t ∈ [ tmin , tmax ] for each patient , given sparse observations y : = ( ỹ ( t1 ) , . . . , ỹ ( tn ) ) . Due to the highdimensional nature of longitudinal data , existing results usually restrict solutions to subspace of functions and utilize similarities between patients via enforcing low-rank structures . One popular approach is the mixed effect models , including Gaussian process approaches ( Verbeke , 1997 ; Zeger et al. , 1988 ) and functional principal components ( James et al. , 2000 ) . While generative models are commonly used and have nice theoretical properties , their result could be sensitive to the underlying distributional assumptions of observed data and hard to adapt to different applications . Another line of research is to pose the problem of disease progression estimation as an optimization problem . Kidzinski and Hastie . Kidziński & Hastie ( 2018 ) proposed a framework which formulates the problem as a matrix completion problem and solve it using matrix factorization techniques . This method is distribution-free and flexible to possible extensions . Meanwhile , both types of solutions model the natural progression of disease using observations of the targeted variables only . They fail to incorporate the existence and effect of human interference : medications , therapies , surgeries , etc . Two patients with similar symptoms initially may have different futures if they choose different treatments . Without that information , predictions can be way-off . To the best of our knowledge , existing literature talks little about modeling treatment effect on disease progression . In Kidziński & Hastie ( 2018 ) , authors use concurrent observations of auxillary variables ( e.g . oxygen consumption to motor functions ) to help estimate the target one , under the assumption that both variables reflect the intrinsic latent feature of the disease and are thus correlated . Treatments of various types , however , rely on human decisions and to some extent , an exogenous variable to the development of disease . Thus they need to modeled differently . In this work , we propose a model for tracking disease progression that includes the effects of treatments . We introduce the Coordinatewise-Soft-Impute ( CSI ) algorithm for fitting the model and investigate its theoretical and practical properties . The contribution of our work is threefold : First , we propose a model and an algorithm CSI , to estimate the progression of disease which incorporates the effect of treatment events . The framework is flexible , distribution-free , simple to implement and generalizable . Second , we prove that CSI converges to the global solution regardless of the initialization . Third , we compare the performance of CSI with various other existing methods on both simulated data and a dataset of Gillette Children ’ s Hospital with patients diagnosed with Cerebral Palsy , and demonstrate the superior performances of CSI . The rest of the paper is organized as follows . In Section 2 we state the problem and review existing methods . Next , in Section 3 we describe the model and the algorithm . Theoretic properties of the algorithm are derived in Section 4 . Finally , in Section 5 and 6 we provides empirical results of CSI on the simulated and the real datesets respectively . We discuss some future directions in Section 7 . 2 PROBLEM STATEMENT AND RELATED WORK . Let y ( t ) be the trajectory of our objective variable , such as the size of tumor , over fixed time range t ∈ [ tmin , tmax ] , and N be the number of patients . For each patient 1 ≤ i ≤ N , we measure its trajectory yi ( t ) at ni irregularly time points ti = [ ti,1 , ti,2 , ... , ti , ni ] ′ and denote the results as yi = [ yi,1 , ... , yi , ni ] ′ = [ yi ( ti,1 ) , ... , yi ( ti , ni ) ] ′ . We are primarily interested in estimating the disease progression trajectories { yi ( t ) } Ni=1 of all N patients , based on observation data { ( ti , yi ) } Ni=1 . To fit a continuous curve based on discrete observations , we restrict our estimations to a finitedimensional space of functions . Let { bi , i ∈ N } be a fixed basis of L2 ( [ tmin , tmax ] ) ( e.g . splines , Fourier basis ) and b = { bi : 1 ≤ i ≤ K } be first K dimensions of it . The problem of estimating yi ( t ) can then be reduced to the problem of estimating the coefficients wi = [ wi,1 , wi,2 , · · · , wi , K ] ′ such that w′ib ( t ) is close to yi ( t ) at time t ∈ ti . Though intuitive , the above method has two main drawbacks . First , when the number of observations per patient is less than or equal to the number of basis functions K , we can perfectly fit any curve without error , leading to overfitting . Moreover , this direct approach ignores the similarities between curves . Different patients may share similar trend of the trajectories which could potentially imporve the prediction . Below we describe two main lines of research improving on this , the mixed-effect model and the matrix completion model . 2.1 LINEAR MIXED-EFFECT MODEL . In mixed-effect models , every trajectory yi ( t ) is assumed to be composed of two parts : the fixed effect µ ( t ) = m′b ( t ) for some m ∈ RK that remains the same among all patients and a random effect wi ∈ RK that differs for each i ∈ { 1 , . . . , N } . In its simplest form , we assume wi ∼ N ( 0 , Σ ) and yi|wi ∼ N ( µi +Biwi , σ2Ini ) , where Σ is the K × K covariance matrix , σ is the standard deviation and µi = [ µ ( ti,1 ) , µ ( ti,2 ) , · · ·µ ( ti , ni ) ] ′ , Bi = [ b ( ti,1 ) , b ( ti,2 ) , · · · , b ( ti , ni ) ] ′ are functions µ ( t ) and b ( t ) evaluated at the times ti , respectively . Estimations of model parameters µ , Σ can be made via expectation maximization ( EM ) algorithm ( Laird & Ware , 1982 ) . Individual coefficients wi can be estimated using the best unbiased linear predictor ( BLUP ) ( Henderson , 1975 ) . In linear mixed-effect model , each trajectory is estimated with |wi| = K degrees of freedom , which can still be too complex when observations are sparse . One typical solution is to assume a low-rank structure of the covariance matrix Σ by introducing a contraction mapping A from the functional basis to a low-dimensional latent space . More specifically , one may rewrite the LMM model as yi|w̃i ∼ N ( µi +BiAw̃i , σ2Ini ) , where A is a K × q matrix with q < K and w̃i ∈ Rq is the new , shorter random effect to be estimated . Methods based on low-rank approximations are widely adopted and applied in practice and different algorithms on fitting the model have been proposed ( James et al. , 2000 ; Lawrence , 2004 ; Schulam & Arora , 2016 ) . In the later sections , we will compare our algorithm with one specific implementation named functional-Principle-Component-Analysis ( fPCA ) ( James et al. , 2000 ) , which uses EM algorithm for estimating model parameters and latent variables wi . 2.2 MATRIX COMPLETION MODEL . While the probabilistic approach of mixed-effect models offers many theoretical advantages including convergence rates and inference testing , it is often sensitive to the assumptions on distributions , some of which are hard to verify in practice . To avoid the potential bias of distributional assumptions in mixed-effect models , Kidzinski and Hastie ( Kidziński & Hastie , 2018 ) formulate the problem as a sparse matrix completion problem . We will review this approach in the current section . To reduce the continuous-time trajectories into matrices , we discretize the time range [ tmin , tmax ] into T equi-distributed points G = [ τ1 , . . . , τT ] with τ1 = tmin , τT = tmax and let B = [ b ( τ1 ) , b ( τ2 ) , · · · , b ( τT ) ] ′ ∈ RT×K be the projection of the K-truncated basis b onto grid G. The N ×K observation matrix Y is constructed from the data { ( ti , yi ) } Ni=1 by rounding the time ti , j of every observation yi ( ti , j ) to the nearest time grid and regarding all other entries as missing values . Due to sparsity , we assume that no two observation yi ( ti , j ) ’ s are mapped to the same entry of Y . Let Ω denote the set of all observed entries of Y . For any matrix A , let PΩ ( A ) be the projection of A onto Ω , i.e . PΩ ( A ) = M where Mi , j = Ai , j for ( i , j ) ∈ Ω and Mi , j = 0 otherwise . Similarly , we define P⊥Ω ( A ) = A− PΩ ( A ) to be the projection on the complement of Ω . Under this setting , the trajectory prediction problem is reduced to the problem of fitting a N ×K matrix W such that WB′ ≈ Y on observed indices Ω . The direct way of estimating W is to solve the optimization problem arg min W 1 2 ‖PΩ ( Y −WB′ ) ‖2F , ( 2.1 ) where ‖ · ‖F is the Fröbenius norm . Again , if K is larger than the number of observations for some subject we will overfit . To avoid this problem we need some additional constraints on W . A typical approach in the matrix completion community is to introduce a nuclear norm penalty—a relaxed version of the rank penalty while preserving convexity ( Rennie & Srebro , 2005 ; Candès & Recht , 2009 ) . The optimization problem with the nuclear norm penalty takes form arg min W 1 2 ‖PΩ ( Y −WB′ ) ‖2F + λ‖W‖∗ , ( 2.2 ) where λ > 0 is the regularization parameter , ‖ · ‖F is the Fröbenius norm , and ‖ · ‖∗ is the nuclear norm , i.e . the sum of singular values . In Kidziński & Hastie ( 2018 ) , a Soft-Longitudinal-Impute ( SLI ) algorithm is proposed to solve ( 2.2 ) efficiently . We refer the readers to Kidziński & Hastie ( 2018 ) for detailed description of SLI while noting that it is also a special case of our algorithm 1 defined in the next section with µ fixed to be 0 .

Summary: The paper introduces Coordinatewise-Soft-Impute (CSI) algorithm that attempts to estimate the progression of a disease while taking into account the effect of a single binary treatment intervention. The authors also provide prove that CSI converges to the global solution regardless of the initialization. The paper is concluded with experimental results on synthetic  and real-world data, comparing the proposed method with contender methods functional Principal Component Analysis (fPCA) and Soft-Longitudinal-Impute (SLI).