This thesis deals with modeling, validation and calibration problems in experiments of computer models. Computer models are mathematic representations of real systems developed for understanding and investigating the systems. Before a computer model is used, it often needs to be validated by comparing the computer outputs with physical observations and calibrated by adjusting internal model parameters in order to improve the agreement between the computer outputs and physical observations. As computer models become more powerful and popular, the complexity of input and output data raises new computational challenges and stimulates the development of novel statistical modeling methods. One challenge is to deal with computer models with random inputs (random effects). This kind of computer models is very common in engineering applications. For example, in a thermal experiment in the Sandia National Lab (Dowding et al. 2008), the volumetric heat capacity and thermal conductivity are random input variables. If input variables are randomly sampled from particular distributions with unknown parameters, the existing methods in the literature are not directly applicable. The reason is that integration over the random variable distribution is needed for the joint likelihood and the integration cannot always be expressed in a closed form. In this research, we propose a new approach which combines the nonlinear mixed effects model and the Gaussian process model (Kriging model). Different model formulations are also studied to have an better understanding of validation and calibration activities by using the thermal problem. Another challenge comes from computer models with functional outputs. While many methods have been developed for modeling computer experiments with single response, the literature on modeling computer experiments with functional response is sketchy. Dimension reduction techniques can be used to overcome the complexity problem of function response; however, they generally involve two steps. Models are first fit at each individual setting of the input to reduce the dimensionality of the functional data. Then the estimated parameters of the models are treated as new responses, which are further modeled for prediction. Alternatively, pointwise models are first constructed at each time point and then functional curves are fit to the parameter estimates obtained from the fitted models. In this research, we first propose a functional regression model to relate functional responses to both design and time variables in one single step. Secondly, we propose a functional kriging model which uses variable selection methods by imposing a penalty function. we show that the proposed model performs better than dimension reduction based approaches and the kriging model without regularization. In addition, non-asymptotic theoretical bounds on the estimation error are presented.

Population PK models aim to describe the change in drug concentration over time for a specific population. The populations in population PK modelling often refer to subjects in a clinical trial of a potential drug candidate. Population PK models are frequently described by non-linear mixed effect (NLME) models, that including both random and fixed effect components. The fixed effect components 𝜽 (THETA) portray typical parameter values in the population while the random effects components 𝜼 (ETA) allow for the incorporation of inter-individual variability (IIV) on the typical population value. The IIVs are therefore an important element of NLME models, but the estimation of the IIVs can be time consuming and become a limiting factor for more complex models. Linear approximation of the IIV’s has been suggested as a way to reduce the estimation time whilst maintaining robustness. The aim of this project was to evaluate and compare the estimation time and robustness of the IIVs for the linear approximation of parameter estimation errors in NLME models compared to those estimated in non-linear models. Population PK NLME models were developed for two datasets of phenobarbital and moxonidine. The datasets contained different levels of complexity such as number of subjects, datapoints and route of administration. The models were developed within R-studio using the assembler and Pharmpy packages and evaluated in NONMEM 7.5. Based on the objective function values (OFVs), obtained in the model building processes, selected models were linearised using Pearl speaks NONMEM (PsN). The estimated 𝜀′𝑠 and run-time of the linearised models were compared to their non-linearized counterparts. For all the models a reduction in run-time could be observed but with a slight variation in the estimations between the linearised and non-linearised models. The biggest run time reduction was seen in the oral transit compartment models for moxonidine with a 3100-fold reduction in estimation time. The estimation time reduction displayed could more quickly provide valuable information regarding the chosen error models of more complex models and while parameters estimated may not be identical to the non-linearised models, they should be sufficient during the model building phase.

Tuberculosis is one the leading causes of death globally and was before the COVID-19 pandemic the leading cause of death from a single infectious agent. Developing active tuberculosis is life threatening and therefore is the rise of drug-resistant tuberculosis alarming as this risk causing current treatments to become ineffective. Linezolid is a promising drug for treatment of drug-resistant pulmonary tuberculosis, but the effect of linezolid treatment for pulmonary tuberculosis subjects is still not understood well enough and the World Health Organization has requested this knowledge gap to be filled. In this project we support the closing of this knowledge gap by describing the pharmacokinetics of linezolid for treatment of pulmonary tuberculosis using data collected from a phase two clinical trial in a South African population. This was done by creating a pop-PK model and resulted in the PK of linezolid in pulmonary tuberculosis patients from South Africa was best described using a one-compartment model, with first-order absorption process preceded by a series of transit compartments and saturable elimination. However, the diagnostics of the model still show that there are room for improvements and future work is necessary to further optimize the model.