Quantifying uncertainty, variability and likelihood for ordinary differential equation models

<p>Abstract</p> <p>Background</p> <p>In many applications, ordinary differential equation (ODE) models are subject to uncertainty or variability in initial conditions and parameters. Both, uncertainty and variability can be quantified in terms of a probability density function on the state and parameter space.</p> <p>Results</p> <p>The partial differential equation that describes the evolution of this probability density function has a form that is particularly amenable to application of the well-known method of characteristics. The value of the density at some point in time is directly accessible by the solution of the original ODE extended by a single extra dimension (for the value of the density). This leads to simple methods for studying uncertainty, variability and likelihood, with significant advantages over more traditional Monte Carlo and related approaches especially when studying regions with low probability.</p> <p>Conclusions</p> <p>While such approaches based on the method of characteristics are common practice in other disciplines, their advantages for the study of biological systems have so far remained unrecognized. Several examples illustrate performance and accuracy of the approach and its limitations.</p

Identifiability and sensitivity analysis of heterogeneous cell population models

In this thesis, we introduce novel concepts to the modeling and analysis of heterogeneous cell populations. Heterogeneous cell populations can be interpreted as large populations of structurally identical cells with heterogeneous parameters and initial conditions. They appear in biological systems such as tissues of higher organisms or colonies of microorganisms. A well-known approach for the modeling of heterogeneous cell populations is the so called density-based approach, in which the state of a heterogeneous cell population is given by the probability density of the cell states. The evolution of the probability densities is in this approach given in terms of a partial differential equation. We extend this approach via a measure theoretical consideration, which exploits the probabilistic nature of the problem. The result of this novel ansatz is a framework in which the evolution of densities is described by operators. One of the key tasks in the analysis of heterogeneous cell population models is parameter estimation. For heterogeneous cell populations we want to estimate the probability density of parameters and initial conditions. However, to be able to perform parameter estimation, one always needs specific identifiability properties of a system. We formulate for the first time the concept of structural identifiability of a heterogeneous cell population model. It is revealed that this concept is closely related to observability of the corresponding single cell model. The connection between both concepts is studied and illuminated in a concrete example. The second emphasis of this thesis is the implementation of sensitivity analysis to the class of heterogeneous cell population models. Here we study sensitivity with respect to variations or misspecifications in the probability density of parameters and initial conditions

ANALYSIS OF BIOPATHWAY MODELS USING PARALLEL ARCHITECTURES

Ph.DDOCTOR OF PHILOSOPH