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

Transcription

1 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 / 52

2 Outline 1 Introduction 2 Statistical estimation and inference methods for dynamic ODE models Naive Method: LS or MLE principle Local solution and time-varying parameter problems Smoothing-based methods Sparse longitudinal data: mixed-effects ODE models Bayesian methods High-dimensional ODE models: ODE model selection 3 Other dynamic models 4 Ongoing and future Work 5 Conclusions Hulin Wu UTSPH March / 52

3 Statistical Modeling Cultures Leo Breiman (Statistical Science, 2001): Two cultures Data modeling (98% statisticians): What the data look like? e.g., regression models Algorithmic modeling (2% statisticians): No models and for prediction purpose, e.g., neural nets and decision trees A third culture: Mechanistic modeling (<1% statisticians): Build mathematical models based on the mechanisms behind the data How are the data generated? Goal: Understand physics principles or biological mechanisms Hulin Wu UTSPH March / 52

4 Dynamic Systems/Models Many engineering and biological systems can be described by dynamic models: Differential equations: Ordinary differential equations (ODE) simplest Delay differential equations (DDE) Hybrid differential equations (HDE) Partial differential equations (PDE) Stochastic differential equations (SDE) Difference equations and state-space models Stochastic processes models: branching process etc. Agent-based models and cellular automata... Hulin Wu UTSPH March / 52

5 Modeling Goals Forward Problems: θ P θ Easier to do Predictions Simulations Inverse Problems: Y θ Θ More challenging Determine model structures/forms Estimate unknown parameters: θ Hulin Wu UTSPH March / 52

6 A Dynamic System: ODE Model where d dt X(t) = G[X(t), θ], X(0) = X 0 (1) Y (t i ) = H[X(t i ), β] + e(t i ), (2) e(t i ) (0, σ 2 I), i = 1,..., n G( ): linear or nonlinear functions H( ): observation functions (θ, β): unknown parameters e(t i ): measurement error The NLS method: n min {Y (t i ) H[X(t i, θ), β]} T {Y (t i ) H[X(t i, θ), β]}, θ,β,x 0 i=1 where X(t i ) evaluated numerically from Eq (1). Hulin Wu UTSPH March / 52

7 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 / 52

8 Identifiability issues Theoretical identifiability: Mathematical identifiability Practical identifiability: Statistical and numerical identifiability Need to be investigated before the inverse problem How to deal with unidentifiable models? Simplify or revise the model Lump some parameters together Fixed some parameters Bayesian approach: Use priors Hulin Wu UTSPH March / 52

9 Identifiability issues: References 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), 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, 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, 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, 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): Hulin Wu UTSPH March / 52

10 Naive NLS Method: Local solution and numerical error problems Local solution problem: Global optimization methods: Differential evolution algorithms and genetic algorithms (Storn et al 1997). Mixture of stochastic global optimization method and deterministic methods: scatter search method (Rodriguez-Fernandez et al. 2006) Numerical error problem: Xue, Miao and Wu (Annals of Statistics, 2010): theoretical results on numerical error vs. measurement error Hulin Wu UTSPH March / 52

11 Naive NLS Method: Time-varying parameter problem Xue, Miao and Wu, Annals of Statistics (2010) dx(t) dt = F {t, X(t), θ, η(t)} The spline approach can be used to approximate the time-varying parameter: η(t) = π(t) T α, where π(t) = (B 1 (t),, B N (t)) T is a vector of basis functions. The time-varying coefficient ODE model becomes an ODE model with constant parameters: dx(t) dt = F {t, X(t), θ, π(t) T α} Hulin Wu UTSPH March / 52

12 Smoothing-Based Approaches: ODE Computational Problem Earlier ideas: Hemker (1972) and Varah (1982) Two-stage decoupling approaches: Chen and Wu (JASA 2008, Statistica Sinica 2008) and Liang and Wu (JASA, 2008) Parameter cascading method: Ramsay et al. JRSS-B (2007) and Wang et al. Stat Comput Hulin Wu UTSPH March / 52

13 Smoothing-Based Approaches: Two-Stage Method Chen and Wu (JASA 2008, Statistica Sinica 2008) and Liang and Wu (JASA, 2008): X (t i ) = F [X(t i ), θ] (3) Y (t i ) = X(t i ) + e 1 (t i ), e 1 (t i ) (0, σ 2 I), (4) Step 1: Use a nonparametric smoothing to estimate X(t) and X (t) from model (4). Step 2: Substitute the estimate ˆX(t i ) into model (3) to obtain: ˆX (t i ) = F [ ˆX(t i ), θ] + e 2 (t i ). (5) Then fit the above regression model (5) to estimate θ. F ( ): Linear or nonlinear function Hulin Wu UTSPH March / 52

14 Smoothing-Based Approaches: Two-Step Methods Step 2 decoupled the system of ODEs: Fit the ODE one-by-one Convert ODE models to regression: Standard regression software tools can be used Avoid numerically solving the ODEs Computationally fast and efficient: Easy to deal with high-dimensional ODEs Price to pay: The derivative estimate may not be accurate The decoupled system: Some information lost The coupled" property: destroyed Extension to higher-order numerical discretization-based algorithms: Wu, Xue and Kuman (Biometrics 2012) Hulin Wu UTSPH March / 52

15 Parameter Cascading or Profiling Method Ramsay, Hooker, Campbell, Cao, JRSS-B, 2007 Fitting to data Observations: y(t i ) Nonparametric function: f(t) = φ(t) c Fitting to data: C 1 = n i=1 [y(t i) f(t i )] 2 Fidelity to DE x (t) = g(x β) f (t) = φ (t)c Difference between two sides of DE: Lf(t) = f (t) g(f(t) β) Fidelity to DE: C 2 = [Lf(t)] 2 dt Criterion to estimate c: J(c β) = C 1 + λc 2 Criterion to estimate β: H(β) = n i=1 [y(t i) φ(t i ) ĉ(β)] 2 Hulin Wu UTSPH March / 52

16 Numerical Comparisons: NLS, Profiling and Two-Stage Estimates Ding and Wu, Statistica Sinica, 2014 NLS: Not stable to get the global solution, computationally expensive Profiling: A 3-step iterative algorithm More stable than NLS to get a better solution Computational efficiency: similar to NLS Two-Stage Method: Computationally fast, but not accurate. Hulin Wu UTSPH March / 52

17 Sparse Longitudinal Data Problem: Mixed-Effects Modeling Approaches Deal with sparse data: Borrow information across subjects The MLE principle: Nonlinear Mixed-Effects Modeling (NLME) Treat the ODE solution as a nonlinear regression function Computational challenge: Stochastic Approximation EM (SAEM) Two-stage smoothing-based mixed-effects modeling approaches Fang, Wu and Zhu, Statistica Sinica (2011) Linear ODE: Linear mixed-effects model (LME) Nonlinear ODE: NLME Bayes methods A three-stage hierarchical model: implemented by MCMC Computation: expensive Hulin Wu UTSPH March / 52

18 Mixed-Effects ODE Model: NLME Within-subject variation: d dt X(t) = G[X(t), θ i], X(0) = X i0 (6) Y i (t i ) = H i [X i (t i ), θ i ] + e i (t i ), i = 1,..., n Xi(t i): ODE solution for Subject i. Y i = (y i1(t 1),, y imi (t mi )) T : Data from Subject i ei = (e i(t 1),, e i(t mi )) T N (0, σ 2 I mi ): Measurement error Between-subject variation: θ i = µ + b i, [b i Σ] N (0, Σ) µ: population parameter bi: random effects Estimation and inference: Stochastic Approximation EM (SAEM) Delyon, Lavielle and Moulines (1999), Kuhn and Lavielle (2005) Grenier, Louvet, Vigneaux (2014) Hulin Wu UTSPH March / 52

19 Smoothing-based Two-Stage Mixed-Effects Model Fang, Wu and Zhu, Statistica Sinica (2011): X (t i ) = F [X(t i ), θ] (7) Y (t i ) = X(t i ) + e 1 (t i ), e 1 (t i ) (0, σ 2 I), (8) Step 1: Use a nonparametric smoothing to estimate X(t) and X (t) from model (8). Step 2: Substitute the estimate ˆX(t i ) into model (7) to obtain: ˆX (t i ) = F [ ˆX(t i ), θ] + e 2 (t i ). (9) Convert the model (9) into a LME or NLME if F (x) is linear or nonlinear. Fit the LME or NLME using a standard approach or SAEM method Hulin Wu UTSPH March / 52

20 Bayesian Methods: Borrow Information to Deal with Sparse Data and Identifiability Problems Huang, Liu and Wu, Biometrics (2006): Example A viral dynamic model: describe the population dynamics of HIV and its target cells in plasma d dt T = λ ρt [1 γ(t)]kt V d dt T = [1 γ(t)]kt V δt d dt V = NδT cv (10) T, T, V : target uninfected cells, infected cells, virus γ(t): time-varying antiviral drug efficacy (λ, ρ, k, δ, N, c): unknown parameters to be estimated The equations (10): no closed-form solution Hulin Wu UTSPH March / 52

21 Antiviral Drug Efficacy Model A modified E max (M-M) model for drug efficacy: γ(t) = C(t)A(t) φic 50 (t) + C(t)A(t) = IQ(t)A(t) φ + IQ(t)A(t), 0 γ(t) 1 C(t): the plasma drug concentration A(t): drug adherence measurements IC50: in vitro phenotype drug resistance marker φ: a conversion factor parameter IQ = C(t) IC 50 : the Inhibitory Quotient (IQ) (t) If γ(t) = 1, the drug: 100% effective If γ(t) = 0, the drug: no effect (11) Hulin Wu UTSPH March / 52

22 Drug Susceptibility Model Phenotype marker IC 50 is used to quantify agent-specific drug sensitivity The function: to describe changes overtime in IC 50 IC 50 (t) = { I0 + Ir I0 t r t for 0 < t < t r, I r for t t r, (12) I0 and I r: respective values of IC 50(t) at baseline and time point t r at which drug resistant mutations appear If Ir = I 0, no resistance mutation developed during treatment Hulin Wu UTSPH March / 52

23 A Challenging Problem How to estimate the unknown parameters in the complex dynamic model? Difficulties: Identifiability problem: Too many parameters, (φ, λ, ρ, k, δ, N, C), some of them are not identifiable Data from individuals: sparse, only V (t) measured Nonlinear differential equations model: no closed-form solutions Hulin Wu UTSPH March / 52

24 Viral load data from a clinical trial Real data up to day 112 Time (days) log10(rna) copies/ml log10(50) Hulin Wu UTSPH March / 52

25 Bayesian Modeling A three-stage Bayesian hierarchical model Stage 1. Within-subject variation: y i = f i(θ i) + e i, [e i σ 2, θ i] N (0, σ 2 I mi ) fi(θ i) = (f i1(θ i, t 1),, f imi (θ i, t mi )) T : ODE solutions. yi = (y i1(t 1),, y imi (t mi )) T : Data from Subject i ei = (e i(t 1),, e i(t mi )) T : Measurement error Stage 2. Between-subject variation: θ i = µ + b i, [b i Σ] N (0, Σ) Stage 3. Hyperprior distributions: σ 2 Ga(a, b), µ N (η, Λ), Σ 1 Wi(Ω, ν) Gamma (Ga), Normal (N ) and Wishart (Wi): independent distributions Hyper-parameters a, b, η, Λ, Ω and ν: known Hulin Wu UTSPH March / 52

26 Bayesian Estimation: Implementation Choose prior distributions Informative prior and non-informative prior Rule of thumb: choose non-informative prior distributions for parameters of interest Implement MCMC algorithm Gibbs sampling step: closed form of conditional distributions for σ 2, µ, Σ 1 Metropolis-Hastings step: no closed form of conditional distributions for θ i Run a long chain: the number of iterations, initial burn-in", every fifth simulation samples Obtain posterior distributions (posterior means or credible intervals) based on the final MCMC samples Hulin Wu UTSPH March / 52

27 A Clinical Study: A5055 A study of HIV-1 infected patients failing PI-containing therapies. Two salvage regimens: 44 patients Arm A: IDV 800 mg q12h+rtv 200mg q12h+two NRTIs Arm B: IDV 400 mg q12h+rtv 400mg q12h+two NRTIs Plasma HIV-1 RNA (viral load) measured at days 0, 7, 14, 28, 56, 84, 112, 140 and 168 of follow-up Hulin Wu UTSPH March / 52

28 Clinical Data Results of Population Parameters Parameter PM SD 95% CI φ (1.2143, ) c (2.7139, ) δ (0.3387, ) λ (91.497, ) ρ (0.0905, ) N ( , ) k ( , ) Posterior mean for the population parameter φ is with a SD of and the 95% CI of (1.2143, ) As φ plays a role of transforming the in vitro IC 50 into in vivo IC 50, our estimate shows that there is about 2-fold difference between in vitro IC 50 and in vivo IC 50 Hulin Wu UTSPH March / 52

29 Clinical Data Results of Individual Parameters Patient φ i c i δ i λ i ρ i N i k i e The individual-specific parameter estimates suggest a large inter-subject variation The model provides a good fit to the clinical data Hulin Wu UTSPH March / 52

30 Patient 1 Fitted individual curves, drug efficacy, IC50 and adherence with IQ=c12h/IC50 Patid= 1 Patid= 1 IC IDV RTV Adherence IDV RTV Time (day) Time (day) Patid= 1 Patid= 1 Drug efficacy ec log10(rna) o o o o o o o o Time (day) Time (day) Hulin Wu UTSPH March / 52

31 Patient 2 Patid= 2 Patid= 2 IC IDV RTV Adherence IDV RTV Time (day) Time (day) Patid= 2 Patid= 2 Drug efficacy ec log10(rna) o o o o o o o o Time (day) Time (day) Hulin Wu UTSPH March / 52

32 Patient 3 Patid= 3 Patid= 3 IC IDV RTV Adherence IDV RTV Time (day) Time (day) Patid= 3 Patid= 3 Drug efficacy ec log10(rna) o o o o o o o o o Time (day) Time (day) Hulin Wu UTSPH March / 52

33 Bayesian Methods: Pros & Cons Pros Use prior to solve the identifiability problem Deal with extremely complicated models such as nonlinear differential equation models Borrow information across subjects: Deal with sparse longitudinal data Estimate parameters for both population and individuals Always get reasonable estimates Use posterior distributions: Easy to quantify uncertainty" for inference Cons Computation: complex and expensive Prior: dominate the results Hulin Wu UTSPH March / 52

34 High-Dimensional ODEs Require computationally fast and efficient methods Need to incorporate variable selection approaches: LASSO, SCAD etc. Easy to deal with longitudinal data: Mixed-effects models Two-stage smoothing-based method: good for this purpose Hulin Wu UTSPH March / 52

35 Linear ODEs Time course gene expression data: Dynamic gene regulatory network (GRN) reconstruction (Lu, Liang, Li and Wu, JASA 2011) dx i dt = n j=1 θ ij x j, i = 1,, n, (13) When n is small, standard statistical inference and variable selection methods can be used When n is large, curse-of-dimensionality Hulin Wu UTSPH March / 52

36 High-Dimensional Linear ODE: Identifying Significant Regulations Two-Stage Method (Chen and Wu 2008a, 2008b; Liang and Wu 2008): Obtain mean expression curves and their derivatives ˆM k (t) and ˆM k (t) from Step II. Substitute ˆM k (t) and ˆM k (t) into the ODE model to form a regression model High Dimensional Linear Regression Model y k (t) = p j=1 β kjx j (t) + ε k (t), k = 1,, p; t = t 1, t 2,..., t N y k (t) = ˆM k (t) and x j(t) = ˆM j (t) Hulin Wu UTSPH March / 52

37 High Dimensional Model Selection Two-stage method Decouple the high-dimensional ODEs Convert the ODE model into a simple linear model Computationally fast Stepwise selection and subset selection Bridge selection (Frank and Friedman 1993) Least absolute shrinkage and selection operator (LASSO) (Tibshirani 1996) Smoothly Clipped Absolute Deviation (SCAD)(Fan and Li 2001; Kim, Choi and Oh 2008) Hulin Wu UTSPH March / 52

38 Estimation Refinement: Stochastic Approximation EM (SAEM) Algorithm Mixed-Effects ODE Model for Module k dx ki dt m k = β kij M [kj] (t), i = 1,, n k ; k = 1,..., p, (14) j=1 Longitudinal Measurement Model y ki (t) = x ki (t) + ε ki (t) (15) Random Effects Model β ki = β k + b ki (16) b ki N (0, D k ) Hulin Wu UTSPH March / 52

39 Application: Identification of Dynamic GRN for Yeast Cell Cycle DNA microarrays experiment: 18 equally spaced time points during two cell cycles (Spellman 1998) Step I: 800 significant genes identified Step II: Cluster 800 genes into 41 functional modules Step III: Smoothing Step IV: Linear ODE model identification: SCAD variable selection Step V: Estimation Refinement Step VI: Function Enrichment Analysis Hulin Wu UTSPH March / 52

40 Yeast Cell Cycle Gene Expression Profile Module Module Module Module Module Module Module Module 8* Module 9* Module 10 Module 11 Module Module 13 Module 14* Module 15* Module Hulin Wu UTSPH March / 52

41 Yeast Cell Cycle Gene Expression Profile Module 17 Module 18* Module 19 Module 20* Module 21 Module 22 Module 23 Module Module 25* Module 26 Module 27 Module Module Module 30* Module Module 32* Hulin Wu UTSPH March / 52

42 Yeast Cell Cycle Gene Expression Profile Module Module Module Module 36 Module 37 Module 38* Module 39 Module Module 41* 2 0 Hulin Wu UTSPH March / 52

43 Graph of Yeast Cell Cycle GRN Hulin Wu UTSPH March / 52

44 High-Dimensional Nonlinear/Nonparametric ODEs Generalized ODEs: Miao, Wu and Xue, Journal of the American Statistical Association (2014) Sparse additive ODEs: Wu, Lu, Xue and Liang, Journal of the American Statistical Association (2014) Additive nonlinear ODEs: Xue, Wu, Wu and Wu, a manuscript (2017) Hulin Wu UTSPH March / 52

45 Other Dynamic Models: State-Space Models (SSM) Linear SSM: where X t+1 = F t X t + V t, V t (0, Q t ) (17) Y t = G t X t + W t, W t (0, R t ) (18) V t and W t : independent model noise and measurement noise Standard Kalman filter (Kalman, 1960): the core algorithm for prediction and smoothing of state state vectors Hulin Wu UTSPH March / 52

46 Statistical Methods for State-Space Models Zhu and Wu, JCGS (2007) Liu, Lu, Niu and Wu, Biometrics (2011) Liu, Wu, Zhu, Miao, BMC Bioinformatics (2014) Chen et al. PlusOne (2017), submitted Hulin Wu UTSPH March / 52

47 Extension to SDE and PDE: Possible but Challenging Theoretically difficult Computationally challenging Applications: Not common Hulin Wu UTSPH March / 52

48 Ongoing and Future Research High-dimensional ODEs: How to improve accuracy without sacrificing too much on computing? Extra-high dimensional ODE: 1000 ODEs with 1 million parameters (Wu, Qiu, Yuan and Wu, 2017, submitted). Characteristic analyses of large ODE systems: Controllability and stability analysis with uncertainty in parameter estimation Sun, Hu, Wu, Qiu, Linel, Wu, Infectious Disease Modelling 2016 AI-driven ODE Model Builder Hulin Wu UTSPH March / 52

49 Conclusions Dynamic Models: Practically useful for both understanding associations and predictions Both theoretically and computationally challenging Statistical methods for dynamic models: More work needed Hulin Wu UTSPH March / 52

50 Dr. Hulin Wu s Publications on ODE Models by Topics ODE identifiability 1. 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), 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, 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, 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): Lee, Y.+ and Wu, H.* (2012), MARS Approach for Global Sensitivity Analysis of Differential Equation Models with Applications to Dynamics of Influenza Infection, Bulletin of Mathematical Biology, 74, Wu, H.*, Miao, H., Xue, H., Topham, D.J., Zand, M. (2015), Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters, Statistics in Biosciences, 7(1): NLS Estimation of ODE parameters 1. Wu, H.*, Huang, Y.+, Dykes, C., Liu, D., Ma, J., Perelson, A.S., Demeter, L. (2006), Modeling and Estimation of Replication Fitness of HIV-1 in Vitro Experiments Using a Growth Competition Assay, Journal of Virology, 80, Xue, H.+, Miao, H., Wu, H.* (2010), Sieve Estimation of Constant and Time-Varying Coefficients in Nonlinear Ordinary Differential Equation Models by Considering Both Numerical Error and Measurement Error, Annals of Statistics, 38(4), Two-stage methods for ODE models 1. Liang, H., Wu, H.*, (2008), Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models, Journal of the American Statistical Association, 103, Chen, J.+ and Wu, H.* (2008), Efficient Local Estimation for Time-varying Coefficients in Deterministic Dynamic Models with Applications to HIV-1 Dynamics, Journal of the American Statistical Association, 103, Fang, Y.+, Wu, H.*, Zhu, L. (2011), A Two-Stage Estimation Method for Random Coefficient Differential Equation Models with Application to Longitudinal HIV Dynamic Data, Statistica Sinica, 21, Lu, T.+, Liang, H., Li, H., Wu, H.* (2011), High Dimensional ODEs Coupled with Mixed- Effects Modeling Techniques for Dynamic Gene Regulatory Network Identification, Journal of the American Statistical Association, 106, Wu, H.*, Xue, H., Kumar A.+ (2012), Numerical Discretization-Based Estimation Methods for Ordinary Differential Equation Models via Penalized Spline Smoothing with Applications in Biomedical Research, Biometrics, 68(2), Ding, A.A. and Wu, H.* (2014), Estimation of ODE Parameters Using Constrained Local Polynomial Regression, Statistica Sinica, 24, Wu, H.*, Miao, H., Xue, H., Topham, D.J., Zand, M. (2015), Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters, Statistics in Biosciences, 7(1):

51 Time-varying parameter estimation in ODE Models 1. Chen, J.+ and Wu, H.* (2008), Efficient Local Estimation for Time-varying Coefficients in Deterministic Dynamic Models with Applications to HIV-1 Dynamics, Journal of the American Statistical Association, 103, Chen, J.+ and Wu, H.* (2008), Estimation of time-varying parameters in deterministic dynamic models, Statistica Sinica, 18, 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, Xue, H.+, Miao, H., Wu, H.* (2010), Sieve Estimation of Constant and Time-Varying Coefficients in Nonlinear Ordinary Differential Equation Models by Considering Both Numerical Error and Measurement Error, Annals of Statistics, 38(4), Cao, J., Huang, J.Z., Wu, H. (2012), Penalized Nonlinear Least Squares Estimation of Time-Varying Parameters in Ordinary Differential Equations, Journal of Computational and Graphical Statistics (JCGS), 21(1), Bayesian and mixed-effects ODE modeling approaches for longitudinal data 1. Wu, H.*, Ding, A. and DeGruttola. V. (1998), Estimation of HIV Dynamic Parameters," Statistics in Medicine, 17, Wu, H.* and Ding, A. (1999), Population HIV-1 Dynamics in Vivo: Applicable Models and Inferential Tools for Virological Data from AIDS Clinical Trials," Biometrics, 55, Huang, Y.+ and Wu, H.* (2006), A Bayesian Approach for Estimating Antiviral Efficacy in HIV Dynamic Models, Journal of Applied Statistics, 33, Huang, Y., Liu, D.+ and Wu, H.* (2006), Hierarchical Bayesian Methods for Estimation of Parameters in a Longitudinal HIV Dynamic System, Biometrics, 62, Fang, Y.+, Wu, H.*, Zhu, L. (2011), A Two-Stage Estimation Method for Random Coefficient Differential Equation Models with Application to Longitudinal HIV Dynamic Data, Statistica Sinica, 21, High-dimensional ODE models and model selections 1. Lu, T.+, Liang, H., Li, H., Wu, H.* (2011), High Dimensional ODEs Coupled with Mixed- Effects Modeling Techniques for Dynamic Gene Regulatory Network Identification, Journal of the American Statistical Association, 106, Wu, H.*, Lu, T.+, Xue, H., and Liang, H. (2014), Sparse Additive ODEs for Dynamic Gene Regulatory Network Modeling, Journal of the American Statistical Association, 109:506, Nonlinear/nonparametric ODE models 1. Wu, H.*, Lu, T.+, Xue, H., and Liang, H. (2014), Sparse Additive ODEs for Dynamic Gene Regulatory Network Modeling, Journal of the American Statistical Association, 109:506, Miao, H., Wu, H., and Xue, H. (2014), Generalized Ordinary Differential Equation Models, Journal of the American Statistical Association, 109:508, Statistical methods for state-space models 1. Zhu, H.+ and Wu, H.* (2007), Estimation of Smoothing Time-Varying Parameters in State Space Models, Journal of Computational and Graphical Statistics (JCGS), 16(4), Liu, D.+, Lu, T.+, Niu, X.F., and Wu, H.* (2011), Mixed-Effects State Space Models for Analysis of Longitudinal Dynamic Systems, Biometrics, 67,

52 ODE experimental design 1. Wu, H.* and Ding, A.A. (2002), Design of Viral Dynamic Studies for Efficiently Assessing Anti-HIV Therapies in AIDS Clinical Trials," Biometrical Journal, 2, Huang, Y. and Wu, H.* (2008), Bayesian Experimental Design for Long-Term Longitudinal HIV Dynamic Studies, Journal of Statistical Planning and Inference, 138, 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): Dynamic model property analysis with uncertainty 1. Sun, X.+, Hu, F.+, Wu, S., Qiu, X., Linel, P.+, Wu, H.* (2016), Controllability and Stability Analysis of Large Transcriptomic Dynamic Systems for Host Response to Influenza Infection in Human, Infectious Disease Modelling, 1(1),

53 Our recent work in nonlinear/high-dimensional ODE models Lu, T., Liang, H., Li, H., Wu, H. (2011), High Dimensional ODEs Coupled with Mixed-Effects Modeling Techniques for Dynamic Gene Regulatory Network Identification, JASA, 106, Wu, H., Xue, H., Kumar A. (2012), Numerical Discretization-Based Estimation Methods for Ordinary Differential Equation Models via Penalized Spline Smoothing with Applications in Biomedical Research, Biometrics, 68(2), Miao, H., Wu, H., and Xue, H. (2014), Generalized Ordinary Differential Equation Models, JASA, 109:508, Wu, H., Lu, T., Xue, H., and Liang, H. (2014), Sparse Additive ODEs for Dynamic Gene Regulatory Network Modeling, JASA, 109:506, Wu, S., Liu, Z.P., Qiu, X., and Wu, H. (2014), Modeling genome-wide dynamic regulatory network in mouse lungs with influenza infection using high-dimensional ordinary differential equations, PLOS ONE, 9(5):e Linel, P., Wu, S., Deng, N., Wu, H. (2014), Dynamic transcriptional signatures and network responses for clinical symptoms in influenza-infected human subjects using systems biology approaches, Journal of PK/PD, 41, Qiu, X. et al. (2015), Diversity in Compartmental Dynamics of Gene Regulatory Networks: The Immune Response in Primary Influenza A Infection in Mice, PLoS ONE, 10(9). Hulin Wu UTSPH March / 52

54 Acknowledgement More than 30 postdocs, students and collaborators Hulin Wu UTSPH March / 52

55 Thank You! Hulin Wu UTSPH March / 52