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Statistical Methods for Bridging Experimental Data and Dynamic Models with Biomedical Applications

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Statistical Methods for Bridging Experimental Data and Dynamic Models with Biomedical Applications Statistical Methods for Bridging Experimental Data and Dynamic Models with Biomedical Applications Hulin Wu, Ph.D. Dr. D.R. Seth Family Professor & Associate Chair Department of Biostatistics, School of Public Health Professor, School of Biomedical Informatics University of Texas Health Science Center at Houston Pittsburgh, March, 2017 Hulin Wu UTSPH March 2017 1 / 52 Outline 1 Introduction 2 Statistical estimation and inference methods for dynamic ODE models I Naive Method: LS or MLE principle I Local solution and time-varying parameter problems I Smoothing-based methods I Sparse longitudinal data: mixed-effects ODE models I Bayesian methods I High-dimensional ODE models: ODE model selection 3 Other dynamic models 4 Ongoing and future Work 5 Conclusions Hulin Wu UTSPH March 2017 2 / 52 Statistical Modeling Cultures I Leo Breiman (Statistical Science, 2001): Two cultures I Data modeling (98% statisticians): What the data look like? e.g., regression models I Algorithmic modeling (2% statisticians): No models and for prediction purpose, e.g., neural nets and decision trees I A third culture: I Mechanistic modeling (<1% statisticians): Build mathematical models based on the mechanisms behind the data I How are the data generated? I Goal: Understand physics principles or biological mechanisms Hulin Wu UTSPH March 2017 3 / 52 Dynamic Systems/Models Many engineering and biological systems can be described by dynamic models: I Differential equations: I Ordinary differential equations (ODE)–simplest I Delay differential equations (DDE) I Hybrid differential equations (HDE) I Partial differential equations (PDE) I Stochastic differential equations (SDE) I Difference equations and state-space models I Stochastic processes models: branching process etc. I Agent-based models and cellular automata I ::: Hulin Wu UTSPH March 2017 4 / 52 Modeling Goals I Forward Problems: θ 7! Pθ–Easier to do I Predictions I Simulations I Inverse Problems: Y 7! θ 2 Θ–More challenging I Determine model structures/forms I Estimate unknown parameters: θ Hulin Wu UTSPH March 2017 5 / 52 A Dynamic System: ODE Model d X(t) = G[X(t); θ]; X(0) = X (1) dt 0 Y (ti) = H[X(ti); β] + e(ti); (2) 2 e(ti) ∼ (0; σ I); i = 1; : : : ; n where I G(·): linear or nonlinear functions I H(·): observation functions I (θ; β): unknown parameters I e(ti): measurement error The NLS method: n X T min fY (ti) − H[X(ti; θ); β]g fY (ti) − H[X(ti; θ); β]g; θ;β;X 0 i=1 where X(ti) evaluated numerically from Eq (1). Hulin Wu UTSPH March 2017 6 / 52 Naive NLS Method: Challenging Problems 1 Identifiability problem 2 Local solutions 3 Time-varying parameters 4 Need to solve the forward problem numerically and many times: Numerical error vs. measurement error 5 Slow convergence and high computational cost 6 Sparse longitudinal data problem 7 Nonlinear optimization 8 High-dimensional parameter space Motivate new statistical methods for dynamic models Hulin Wu UTSPH March 2017 7 / 52 Identifiability issues I Theoretical identifiability: Mathematical identifiability I Practical identifiability: Statistical and numerical identifiability I Need to be investigated before the inverse problem I How to deal with unidentifiable models? I Simplify or revise the model I Lump some parameters together I Fixed some parameters I Bayesian approach: Use priors Hulin Wu UTSPH March 2017 8 / 52 Identifiability issues: References I Wu, H., Zhu, H., Miao, H., and Perelson, A.S. (2008), Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models, Bulletin of Mathematical Biology, 70(3), 785-799. I Miao, H., Dykes, C., Demeter, L.M., Cavenaugh, J., Park, S.Y., Perelson, A.S., and Wu, H. (2008), Modeling and Estimation of Kinetic Parameters and Replicative Fitness of HIV-1 from Flow-Cytometry-Based Growth Competition Experiments, Bulletin of Mathematical Biology, 70, 1749-1771. I Miao, H., Dykes, C., Demeter, L., Wu, H. (2009), Differential Equation Modeling of HIV Viral Fitness Experiments: Model Identification, Model Selection, and Multi-Model Inference, Biometrics, 65, 292-300. I Liang, H., Miao, H., and Wu, H. (2010), Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model, Annals of Applied Statistics, 4, 460-483. I Miao, H., Xia, X., Perelson, A.S., Wu, H. (2011), On Identifiability of Nonlinear ODE Models and Applications in Viral Dynamics, SIAM Review, 53(1): 3-39. Hulin Wu UTSPH March 2017 9 / 52 Naive NLS Method: Local solution and numerical error problems I Local solution problem: I Global optimization methods: Differential evolution algorithms and genetic algorithms (Storn et al 1997). I Mixture of stochastic global optimization method and deterministic methods: scatter search method (Rodriguez-Fernandez et al. 2006) I Numerical error problem: I Xue, Miao and Wu (Annals of Statistics, 2010): theoretical results on numerical error vs. measurement error Hulin Wu UTSPH March 2017 10 / 52 Naive NLS Method: Time-varying parameter problem Xue, Miao and Wu, Annals of Statistics (2010) dX(t) = F ft; X(t); θ; η(t)g dt I The spline approach can be used to approximate the time-varying parameter: η(t) = π(t)T α; T where π(t) = (B1(t); ··· ;BN (t)) is a vector of basis functions. I The time-varying coefficient ODE model becomes an ODE model with constant parameters: dX(t) = F ft; X(t); θ; π(t)T αg dt Hulin Wu UTSPH March 2017 11 / 52 Smoothing-Based Approaches: ODE Computational Problem I Earlier ideas: Hemker (1972) and Varah (1982) I Two-stage decoupling approaches: Chen and Wu (JASA 2008, Statistica Sinica 2008) and Liang and Wu (JASA, 2008) I Parameter cascading method: Ramsay et al. JRSS-B (2007) and Wang et al. Stat Comput 2014. Hulin Wu UTSPH March 2017 12 / 52 Smoothing-Based Approaches: Two-Stage Method Chen and Wu (JASA 2008, Statistica Sinica 2008) and Liang and Wu (JASA, 2008): 0 X (ti) = F [X(ti); θ] (3) 2 Y (ti) = X(ti) + e1(ti); e1(ti) ∼ (0; σ I); (4) I Step 1: Use a nonparametric smoothing to estimate X(t) and X0(t) from model (4). I Step 2: Substitute the estimate X^(ti) into model (3) to obtain: 0 X^ (ti) = F [X^(ti); θ] + e2(ti): (5) Then fit the above regression model (5) to estimate θ. I F (·): Linear or nonlinear function Hulin Wu UTSPH March 2017 13 / 52 Smoothing-Based Approaches: Two-Step Methods I Step 2 decoupled the system of ODEs: Fit the ODE one-by-one I Convert ODE models to regression: Standard regression software tools can be used I Avoid numerically solving the ODEs I Computationally fast and efficient: Easy to deal with high-dimensional ODEs I Price to pay: I The derivative estimate may not be accurate I The decoupled system: Some information lost I The “coupled" property: destroyed Extension to higher-order numerical discretization-based algorithms: Wu, Xue and Kuman (Biometrics 2012) Hulin Wu UTSPH March 2017 14 / 52 Parameter Cascading or Profiling Method Ramsay, Hooker, Campbell, Cao, JRSS-B, 2007 Fitting to data I Observations: y(ti) 0 I Nonparametric function: f(t) = φ(t) c Pn 2 I Fitting to data: C1 = i=1[y(ti) − f(ti)] Fidelity to DE x0(t) = g(xjβ) 0 0 I f (t) = φ (t)c 0 I Difference between two sides of DE: Lf(t) = f (t) − g(f(t)jβ) R 2 I Fidelity to DE: C2 = [Lf(t)] dt Criterion to estimate c: J(cjβ) = C1 + λC2 Pn 0 2 Criterion to estimate β: H(β) = i=1[y(ti) − φ(ti) c^(β)] Hulin Wu UTSPH March 2017 15 / 52 Numerical Comparisons: NLS, Profiling and Two-Stage Estimates Ding and Wu, Statistica Sinica, 2014 I NLS: Not stable to get the global solution, computationally expensive I Profiling: I A 3-step iterative algorithm I More stable than NLS to get a better solution I Computational efficiency: similar to NLS I Two-Stage Method: Computationally fast, but not accurate. Hulin Wu UTSPH March 2017 16 / 52 Sparse Longitudinal Data Problem: Mixed-Effects Modeling Approaches Deal with sparse data: Borrow information across subjects I The MLE principle: Nonlinear Mixed-Effects Modeling (NLME) I Treat the ODE solution as a nonlinear regression function I Computational challenge: Stochastic Approximation EM (SAEM) I Two-stage smoothing-based mixed-effects modeling approaches I Fang, Wu and Zhu, Statistica Sinica (2011) I Linear ODE: Linear mixed-effects model (LME) I Nonlinear ODE: NLME I Bayes methods I A three-stage hierarchical model: implemented by MCMC I Computation: expensive Hulin Wu UTSPH March 2017 17 / 52 Mixed-Effects ODE Model: NLME I Within-subject variation: d X(t) = G[X(t); θ ]; X(0) = X (6) dt i i0 Y i(ti) = Hi[Xi(ti); θi] + ei(ti); i = 1; : : : ; n I Xi(ti): ODE solution for Subject i. T I Y i = (yi1(t1); ··· ; yimi (tmi )) : Data from Subject i T 2 I ei = (ei(t1); ··· ; ei(tmi )) ∼ N (0; σ Imi ): Measurement error I Between-subject variation: θi = µ + bi; [bijΣ] ∼ N (0; Σ) I µ: population parameter I bi: random effects I Estimation and inference: Stochastic Approximation EM (SAEM) I Delyon, Lavielle and Moulines (1999), Kuhn and Lavielle (2005) Grenier, Louvet, Vigneaux (2014) Hulin Wu UTSPH March 2017 18 / 52 Smoothing-based Two-Stage Mixed-Effects Model Fang, Wu and Zhu, Statistica Sinica (2011): 0 X (ti) = F [X(ti); θ] (7) 2 Y (ti) = X(ti) + e1(ti); e1(ti) ∼ (0; σ I); (8) I Step 1: Use a nonparametric smoothing to estimate X(t) and X0(t) from model (8). I Step 2: Substitute the estimate X^(ti) into model (7) to obtain: 0 X^ (ti) = F [X^(ti); θ] + e2(ti): (9) I Convert the model (9) into a LME or NLME if F (x) is linear or nonlinear.
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