@PHDTHESIS\{IMM2006-04677,
    author       = "R. V. Overgaard",
    title        = "Pharmacokinetic/Pharmacodynamic modelling with a stochastic perspective. Insulin secretion and Interleukin-21 development as case studies",
    year         = "2006",
    school       = "Informatics and Mathematical Modelling, Technical University of Denmark, {DTU}",
    address      = "Richard Petersens Plads, Building 321, {DK-}2800 Kgs. Lyngby",
    type         = "",
    note         = "Supervised by Prof. Henrik Madsen, {IMM}.",
    url          = "http://www2.compute.dtu.dk/pubdb/pubs/4677-full.html",
    abstract     = "Mathematical models are used for many different purposes during the development of new drugs. These models can for example help to figure out how much medicine should be given and how often it should be given in order to obtain the desired effect. In other words, the models can help to create the users manual for a new medicinal product. These models are called Pharmacokinetic/Pharmacodynamic (PK/PD) models, where the {PK} part typically describes the concentration of drug in the body, and the {PD} part describes the effect of the drug. If one, for example, develop a {PK}/{PD} model for aspirin, the {PK} part could describe the concentration of aspirin in the blood after you take the tablet. Initially the concentration will increase gradually, and at some point the concentration will begin to decline. The {PD} part could for example describe the level of pain that start of high, begin to decrease after you take the tablet, and presumably increase again when the amount of aspirin has been eliminated from the body. The typical {PK}/{PD} model can be created based on data from an existing experiment, e.g. measurements of concentration and pain relieve at various time points. These models can then simulate the results of new experiments and thereby quickly and inexpensively investigate, e.g. whether aspirin should be administered two or three times daily to obtain the desired effect. The results will naturally be tested in new experiments before the users manual can be accepted for a new medicinal product.

In the present project, new methods are investigated for the formulation and estimation of these mathematical {PK}/{PD} models. Specifically, it is investigated whether stochastic differential equations (SDEs) may improve {PK}/{PD} models and {PK}/{PD} model results. SDEs can be understood as differential equations where the solution is not completely predictable. This randomness could occur, e.g. if there are random variations in the speed with which the drug is removed from the body. In our previous example, one could imagine that this would lead to small fluctuations in the concentration of aspirin in the blood. Biological systems in general are often composed of numerous sub-processes that cannot be expected to perform completely identical from occasion to occasion or from minute to minute. In this way random fluctuations can occur, also because of perturbations from processes that are not modelled, and it is argued that SDEs provide a more natural description of these systems than ordinary differential equations.

During the course of the present project, several models with many different purposes have been developed. These models are developed within two main subjects, insulin secretion and development of {IL-}21 as a new anti cancer drug. We find that SDEs are useful in many aspects of {PK}/{PD} modelling, both for insulin secretion modelling and for models used during the development of {IL-}21. Most importantly, SDEs could improve the models ability to execute their respective main purposes, to describe, predict, or increase the understanding of the system."
}