Non linear models for mechanistic investigation and effect prediction

Team: Tim Holland-LetzAnnette Kopp-Schneider (former members Xiaoqi Jiang, Alla Slynko)  

Mechanistic models for cell population dynamics

Mechanistic models describe the fate of individual cells of different types. In these models, cells may replicate, die or transform to a different type. Small cell populations are described by stochastic models, and the description of events occurring for every single cell are usually the basis for Markovian models. Large populations can be described by deterministic models involving systems of differential equations. One aim of modeling is to investigate biological mechanisms by translating different biological hypotheses into different mathematical models. A comparison of the fit of the models to real data allows us to distinguish between the mechanisms represented in the models.

Models investigated so far are models for carcinogenesis, which describe the process of transformation of normal to preneoplastic and preneoplastic to malignant cells, as well as the proliferation of all cell types. Accordingly, the data needed for model fitting are the number and sizes of premalignant and malignant clones. More recently, models have been developed to describe the effect of neurotoxic substances on different types of brain cells. Brain cells respond to the exposure to neurotoxic substances by transforming to a stressed or activated state and eventually dying as a consequence of intoxication. In immunology, models have been developed that describe the differentiation process of different types of cells in the blood system. Here, transition speed and transition probabilities between different types of blood cells, from long-term stem cells to the different kinds of red and white blood cells, are described. The differentiation process of thymic epithelial cells is another subject of mathematical modeling.

Dose response models

When cell systems are exposed to substances or other treatments, the rates at which changes occur in a system can be described as functions of the dose of exposure. Using the dose dependency of the rates leads to biologically based mechanistic dose-response models. In the end, these models can be employed to predict the effect of the dose on the cell system, especially doses for which experimental data are difficult to obtain. Biologically based dose-response models can also be utilized to compare the biological effect of different exposures on the cell system and hence to elucidate the biological mechanism of the exposure.

In addition to biologically based mechanistic models, statistical models are available for curve-fitting in dose-response experiments. In contrast to biologically based models, statistical models are implemented as nonlinear regression models in standard statistical software packages and are easily applied to data. They do not allow for biological interpretation of the effect of exposure on parameters at the cellular level. However, statistical models include parameters that are used for a more global characterization of exposure, such as the EC50, the dose, or the concentration leading to the half-maximal effect on tissue. Typical models are the log-logistic, the log-normal and the Weibull models.

Current research in statistical dose-response modeling addresses the analysis of synergistic effects in dose-response studies with multiple exposures. Concerning the design of dose-response studies, current research covers the application and extension of mathematical optimal design theory to nonlinear dose-response models. The aim is the identification of those experimental dose levels that guarantee the optimal precision of all desired estimates in a specific trial. In this way, the number of experimental units required for the study can be minimized.


Mechanistic modeling

  • Groos J, Kopp-Schneider A. (2010) Application of a two-phenotype color-shift model with heterogeneous growth to a rat hepatocarcinogenesis experiment. Mathematical Biosciences 224, 95-100.
  • Kopp-Schneider, A. (1997) Carcinogenesis models for risk assessment. Statistical Methods in Medical Research 6, 317-340.
  • Renner, M., Zurich, M.-G., Kopp-Schneider, A. (2013) Stochastic time-concentration activity models for cytotoxicity in 3D brain cell cultures. Theoretical Biology and Medical Modelling 10:19, doi: 10.1186/1742-4682-10-19.

Dose response models

  • Weimer M, Jiang X, Ponta O, Stanzel S, Freyberger A., Kopp-Schneider A. (2012) The impact of data transformations on concentration-response modeling. Toxicology Letters 213, 292– 298.
  • Jiang X, Kopp-Schneider A. Summarizing EC50 estimates from multiple dose-response experiments: a comparison of a meta-analysis strategy to a mixed-effects model approach. Biometrical Journal, doi: 10.1002/bimj.201300123.
  • Holland-Letz, T., Dette, H, Renard, D (2012): Efficient algorithms for optimal designs with correlated observations in pharmacokinetics and dose finding studies, Biometrics 68, 138-145.

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