An algorithm for proper lumping of systems of ODEs
Unviersity of Manchester
Objectives: We aim to develop an algorithm for automatic order reduction of large mathematical models, in order to use it for simplifying systems biology models with drug interest, for potential use in pharmacodynamics.
Methods: We develop an optimisation based algorithm for lumping of responses. More specifically, given a non-linear model described by a set of ordinary differential equations, a set of prior distributions for the model parameters and a set of constraints, the algorithm first determines all the candidate lumped models based on permutations. Then it searches for the one that has the best average agreement with the original model according to a Bayesian objective function that averages over the parameter prior distributions. The algorithm is applied to an example model in order to demonstrate its performance.
Results: The algorithm when applied to the example model, produces a lumped model which is much smaller to the original. Furthermore, the variables and parameters of the reduced model retain a specific physiological meaning, since the algorithm only considers proper lumping schemes. The solution of the lumped model is found to be close to the one of the original full model. Advantages of the algorithm include that it is completely automatic, can be used for non-linear models and can handle parameter uncertainty and constraints.
Conclusions: In the future more sophisticated, mechanistic models will be needed to meet the challenges of quantitative pharmacology. Systems biology is a growing trend in the entire biological sciences and PK/PD modelling needs to go towards that direction also. The inherent problem of overparametrisation of the models derived by a systems approach, given the quality of the available data, will have to be addressed with mathematical tools like lumping techniques, in order for these models to be useful.