Dynamic Bayesian sensitivity analysis of a myocardial metabolic model

Dynamic compartmentalized metabolic models are identified by a large number of parameters, several of which are either non-physical or extremely difficult to measure. Typically, the available data and prior information is insufficient to fully identify the system. Since the models are used to predict the behavior of unobserved quantities, it is important to understand how sensitive the output of the system is to perturbations in the poorly identifiable parameters. Classically, it is the goal of sensitivity analysis to asses how much the output changes as a function of the parameters. In the case of dynamic models, the output is a function of time and therefore its sensitivity is a time dependent function. If the output is a differentiable function of the parameters, the sensitivity at one time instance can be computed from its partial derivatives with respect to the parameters. The time course of these partial derivatives describes how the sensitivity varies in time. When the model is not uniquely identifiable, or if the solution of the parameter identification problem is known only approximately, we may have not one, but a distribution of possible parameter values. This is always the case when the parameter identification problem is solved in a statistical framework. In that setting, the proper way to perform sensitivity analysis is to not rely on the values of the sensitivity functions corresponding to a single model, but to consider the distributed nature of the sensitivity functions, inherited from the distribution of the vector of the model parameters. In this paper we propose a methodology for analyzing the sensitivity of dynamic metabolic models which takes into account the variability of the sensitivity over time and across a sample. More specifically, we draw a representative sample from the posterior density of the vector of model parameters, viewed as a random variable. To interpret the output of this doubly varying sensitivity analysis, we propose visualization modalities particularly effective at displaying simultaneously variations over time and across a sample. We perform an analysis of the sensitivity of the concentrations of lactate and glycogen in cytosol, and of ATP, ADP, NAD(+) and NADH in cytosol and mitochondria, to the parameters identifying a three compartment model for myocardial metabolism during ischemia.