Nonlinear Mixed Effects Modeling Department of Mathematics Center for Research in Scientific Computation Center for Quantitative Sciences in Biomedicine North Carolina State University July 31, 216
Introduction Application of a mathematical model to longitudinal data requires that it is placed in a framework to acknowledge intra-subject variation measurement/assay errors model misspecification/numerical approximation Application of a mathematical model to data from multiple subjects requires that it is placed in a framework that not only recognizes within-subject variation, as above, but also inter-subject variation variation in dynamic parameters across the population (heterogeneity of subjects within the population)
An Example: Logistic Equation Our test model is the logistic equation dp dt = rp ( 1 P K The logistic equation (Verhulst model [1847]) is used to model population growth )
For our test case, data was generated using P() = 2, t =, 1, 2,..., 1 v k N(,.1) To create variability, 1 profiles were created with different parameters having a normal distribution with mean and covariance ( ) ( ) ( ) r.725.5 µ = = Ω = K 2.2
Data 6 Population growth data for 1 groups 5 Population size 4 3 2 1 2 4 6 8 1 time
Two Stage Method Two Stage Method Fit each individual profile find mean of estimates: µ = 1 N N i=1 q i find covariance of estimates: Cov(x, y) = E[xy] µ x µ y Fits Population size 25 2 15 1 Model fits for two groups 5 2 4 6 8 1 time
Two stage fits 6 Two stage population fits 5 Population size 4 3 2 1 2 4 6 8 1 time
True ( ( ) r.725 = K) 2 ( ).5 Ω =.2 Two Stage Results ( ) ( ) r.7551 = K 23.168 ( ).56.14 Ω =.14.2552
Can we do better?
Nonlinear mixed effects models (Davidian/Giltinan95; Wu/Zhang6) are statistical models that are used to analyze repeated measure data, and a modeling framework involving both fixed-effects for population parameters and random effects incorporating uncertainty associated with inter- and intra-individual variability. Stage 1:(intra-individual variability - assay errors, model errors) y i,j = g(x i,j, φ i ) + e i,j, e i,j N(, σ 2 ) i = 1,..., N, j = 1,..., n i dx dt = f (t, x, u; φ), x() = x Stage 2: (inter-individual variability) φ i = h(θ, Z i, η i ); (log-normal ) φ i = θ exp(η i ), η i N(, Ω) Estimate population parameters: (θ, σ 2, Ω)
Maximum Likelihood Estimation Maximum likelihood estimation is a method to estimate parameters (θ, σ 2, Ω) in a statistical model. Maximizing likelihood maximizes the probability of the observed data under the resulting distribution. Maximize the marginal density N L(θ, σ 2, Ω) p 1 (Y ini η i, θ, σ 2 )p 2 (η i Ω)dη i i=1 where L is the population likelihood function, Y ij = [y i1,..., y ini ] represents all observations of the ith individual up to time t ij.
N L(θ, σ 2, Ω) p 1 (Y ini η i, θ, σ 2, d i )p 2 (η i Ω)dη i i=1 Assuming a normal conditional density, the first stage distribution is p 1 (y ini η i, θ, σ 2, d i ) n i j=1 exp( 1 2 et i,j R 1 i(j j 1) e i,j) 2πRi(j j 1) where e i,j = y i,j g(x i,j, φ i ) and R i(j j 1) = prediction covariance. The second stage distribution is p 2 (η i Ω) N(, Ω),
N L(θ, σ 2, Ω) = i=1 N i=1 p 1 (Y ini η i, θ, σ 2, d i )p 2 (η i Ω)dη i exp(l i )dη i, a posteriori log-likehood function for a random effect of the ith individual l i = 1 2 n i j=1 ( ) eij T R 1 i(j j 1) e ij + log 2πR i(j j 1) 1 2 ηt i Ω 1 η i 1 2 log 2πΩ
Logistic Equation True vs. NLME ( ) ( ) r.725 = vs. K 2 ( ( ) r.735 = K) NLME 2.46 ( ).5 Ω = vs..2 ( ).55.18 Ω NLME =.18.255 Fits Population size 6 5 4 3 2 1 NLME population fits 2 4 6 8 1 time
Population Pharmacokinetic Study of Metformin Metformin is a commonly prescribed treatment for type 2 diabetes, with a poorly understood glucose-lowering action. A 16 subject study of a new 2mg extended release (XR) single dose oral formulation of metformin was carried out and plasma concentrations were collected over a 36 hour time frame (every half hour (first 3 hours and between hours 12 to 14); otherwise, every hour).
A Two Compartment Oral Absorption Model A simple two compartment model is used to describe the pharmacokinetics model of the single dose oral administration of Metformin.
Two Compartment Oral Absorption - Model d dt A a C A peripheral with observation equation k a = k a k V (k 12 + k e ) 21 V k 12 V k 21 log(c obs ) = log(c) + e, e N(, σ 2 ) Individual parameter vector (rate constants) φ i = θ exp(η i ), η i N(, Ω) A a C A peripheral
NLME population PK Concentration (mg/l) Predicted Conc. 2 2 Cpt Oral log additive 4 1 2 3 time (h) Invidiual Predictions vs. Observations in log space 1 1 2 2 1.5 1.5.5 1 Observed Conc.
How do we refine the model? Decompose the intra-individual variability into two components: model misspecification term representing the uncertainty associated with unknown or incorrectly specified dynamics measurement noise term dx = f (x, u, t, φ)dt + σdw Nonlinear Filtering
NLME - revisited The population likelihood N L(θ, σ 2, Ω) i=1 p 1 (Y ini η i, θ, σ 2, d i )p 2 (η i Ω)dη i where the first stage distribution is p 1 (y ini η i, θ, σ 2, d i ) n i j=1 Filtering Integration: (Overgaard5) exp( 1 2 et i,j R 1 i(j j 1) e i,j) 2πRi(j j 1) R i(j j 1) = H ij P i(j j 1) H T ij + σ 2 ŷ i(j j 1) = g( x i(j j 1), φ i ),
Two-Compartment Absorption Model- revisited Due to variable absorption, use a stochastic differential equation (SDE) model k a d A a C = k a A a k a V A a (k 12 + k e )C + k21 V A dt peri A peri k 12 VC k 21 A peri σ ka + σ Aa σ central σ peri dw t
SDE Model Calibration Concentration (mg/l Predicted EKF Conc. 2 Cpt Oral log additive EKF 2 4 1 2 3 time (h) Invidiual EKF Predictions vs. Observations in log space 1 1 2 2 1.5 1.5.5 1 Observed Conc. Concentration (mg/l Predicted UKF Conc. 2 Cpt Oral log additive UKF 2 4 1 2 3 time (h) Invidiual UKF Predictions vs. Observations in log space 1 1 2 2 1.5 1.5.5 1 Observed Conc. Figure: SDE Model Fit: EKF Figure: SDE Model Fit: UKF
A Structured Model for Absorption Rate Utilizing information from the SDE model, a structural model for absorption rate (Wiebull) is considered: k a = α ( 1 exp( (λ/t) K ) ) d dt A a C A peripheral k a = k a k V (k 12 + k e ) 21 V k 12 V k 21 A a C A peripheral
Wiebull Model Calibration Concentration (mg/l) Predicted Conc. 2 Cpt Oral log additive Wiebull 2 4 1 2 3 time (h) Invidiual Predictions vs. Observations in log space 1 1 2 2 1.5 1.5.5 1 Observed Conc.
SDE Model Calibration Conc (mg/l).5.5 1 1.5 2 2.5 3 Subject 1 data UKF +σ σ EKF +σ σ 3.5 4 5 1 15 2 25 time (h) Figure: Subject one fits Figure: EKF vs. UKF prediction intervals
Conclusion Noise in Biology can present itself in many ways and the proper handling of this noise is important for both the methodologies and the modeling process. The use of nonlinear filters for the estimation of both state and parameters has shown encouraging results and presents advantages to classical approaches (ODE) in the context of nonlinear mixed effects model.