Accelerating delayed-acceptance Markov chain Monte Carlo algorithms

Delayed-acceptance Markov chain Monte Carlo (DA-MCMC) samples from a probability distribution via a two-stages version of the Metropolis-Hastings algorithm, by combining the target distribution with a "surrogate" (i.e. an approximate and computationally cheaper version) of said distribution. DA-MCMC accelerates MCMC sampling in complex applications, while still targeting the exact distribution. We design a computationally faster, albeit approximate, DA-MCMC algorithm. We consider parameter inference in a Bayesian setting where a surrogate likelihood function is introduced in the delayed-acceptance scheme. When the evaluation of the likelihood function is computationally intensive, our scheme produces a 2-4 times speed-up, compared to standard DA-MCMC. However, the acceleration is highly problem dependent. Inference results for the standard delayed-acceptance algorithm and our approximated version are similar, indicating that our algorithm can return reliable Bayesian inference. As a computationally intensive case study, we introduce a novel stochastic differential equation model for protein folding data.Comment: 40 pages, 21 figures, 10 table

Modelling of drug-effect on time-varying biomarkers

Model-based quantification of drug effect is an efficient tool during pre-clinicaland clinical phases of drug trials. Mathematical modelling can lead to improvedunderstanding of the underlying biological mechanisms, help in finding shortcomingsof experimental design and suggest improvements, or be an effective toolin simulation-based analyses. This thesis addresses the modelling of time-varyingbiomarkers both with and without drug-treatment. Pharmacokinetic/pharmacodynamicmodels were used to describe observed drug concentrations and biomarkers.These are modelled in the framework of compartmental modelling described byordinary differential equations.This thesis contains two papers in manuscript-form. In the first paper, a metaanalysiswas performed of an existing model and previously published data for thestress-hormone cortisol and the drug dexamethasone. Cortisol exhibits a circadianrhythm, resembling oscillations, and is therefore a time-varying target for treatment.The aim was to utilize the model for prediction of the outcome of a medicaltest used in veterinary treatments on horses. In addition to model parameters,inter-individual variability was modelled and estimated in a Bayesian framework.This allowed simulation of test outcomes for the whole population, which in turnwere used to evaluate available test protocols and suggest improvements.In the second paper, an improved model was constructed for the cytokine TNFαafter challenge with LPS in addition to intervention treatment. TNFα is not measurablein healthy subjects but release into blood plasma can be provoked bychallenge with LPS. The result is a short-lived turnover of TNFα. A test compoundtargeting intervention of TNFα release was included in the study. Comprehensiveexperimental data from two studies was available and allowed to model features ofTNFα release, that were not addressed in previously published models. The finalmodel was then used to analyse the current experimental design and correlationsbetween LPS challenge and test compound effectiveness. The paper providessuggestions for future experimental designs

Mixed Effects Modeling of Deterministic and Stochastic Dynamical Systems - Methods and Applications in Drug Development

Mathematical models based on ordinary differential equations (ODEs) are commonly used for describing the evolution of a system over time. In drug development, pharmacokinetic (PK) and pharmacodynamic (PD) models are used to characterize the exposure and effect of drugs. When developing mathematical models, an important step is to infer model parameters from experimental data. This can be a challenging problem, and the methods used need to be efficient and robust for the modeling to be successful. This thesis presents the development of a set of novel methods for mathematical modeling of dynamical systems and their application to PK-PD modeling in drug development.A method for regularizing the parameter estimation problem for dynamical systems is presented. The method is based on an extension of ODEs to stochastic differential equations (SDEs), which allows for stochasticity in the system dynamics, and is shown to lead to a parameter estimation problem that is easier to solve.The combination of parameter variability and SDEs are investigated, allowing for an additional source of variability compared to the standard nonlinear mixed effects (NLME) model. For NLME models with dynamics described using either ODEs or SDEs, a novel parameter estimation algorithm is presented. The method is a gradient-based optimization method where the exact gradient of the likelihood function is calculated using sensitivity equations, which is shown to give a substantial improvement in computational speed compared to existing methods. The methods developed have been integrated into NLMEModeling, a freely available software package for mixed effects modeling in Wolfram Mathematica. The package allows for general model specifications and offers a user-friendly environment for NLME modeling of dynamical systems.The SDE-NLME framework is used in two applied modeling problems in drug development. First, a previously published PK model of nicotinic acid is extended to incorporate SDEs. By extending the ODE model to an SDE model, it is shown that an additional source of variability can be quantified. Second, the SDE-NLME framework is applied in a model-based analysis of peak expiratory flow (PEF) diary data from two Phase III studies in asthma. The established PEF model can describe several aspects of the PEF dynamics, including long-term fluctuations. The association to exacerbation risk is investigated using a repeated time-to-event model, and several characteristics of the PEF dynamics are shown to be associated with exacerbation risk.The research presented in this doctoral thesis demonstrates the development of a set of methods and applications of mathematical modeling of dynamical systems. In this work, the methods were primarily applied in the field of PK-PD modeling, but are also applicable in other scientific fields