A practical approach to parameter estimation applied to model predicting heart rate regulation

Abstract

Mathematical models have long been used for prediction of dynamics in biological systems. Recently, several efforts have been made to render these models patient specific. One way to do so is to employ techniques to estimate parameters that enable model based prediction of observed quantities. Knowledge of variation in parameters within and between groups of subjects have potential to provide insight into biological function. Often it is not possible to estimate all parameters in a given model, in particular if the model is complex and the data is sparse. However, it may be possible to estimate a subset of model parameters reducing the complexity of the problem. In this study, we compare three methods that allow identification of parameter subsets that can be estimated given a model and a set of data. These methods will be used to estimate patient specific parameters in a model predicting baroreceptor feedback regulation of heart rate during head-up tilt. The three methods include: structured analysis of the correlation matrix, analysis via singular value decomposition followed by QR factorization, and identification of the subspace closest to the one spanned by eigenvectors of the model Hessian. Results showed that all three methods facilitate identification of a parameter subset. The "best" subset was obtained using the structured correlation method, though this method was also the most computationally intensive. Subsets obtained using the other two methods were easier to compute, but analysis revealed that the final subsets contained correlated parameters. In conclusion, to avoid lengthy computations, these three methods may be combined for efficient identification of parameter subsets.

Fig. 1

Afferent baroreceptor nerves in the carotid sinus and the aorta are stimulated in response to changes in stretch of the arterial wall, and consequently by changes in arterial pressure. Afferent signals are integrated in the NTS, where sympathetic and parasympathetic efferent signals are generated. These travel to the heart (among other organs), where heart rate is either stimulating or inhibited.

Fig. 2

Model components. Mean blood pressure, age and resting heart rate is used as an input to the model. The model consists of 5 components: a module predicting averaged arterial blood pressure, a module predicting the baroreflex firing rate, modules predicting sympathetic and parasympathetic outflow, and a heart rate module. Dependencies between modules are marked by arrows.

Fig. 3

The figure shows blood pressure (blue), heart rate data (red), and simulated heart rate (cyan) as a function of time.

Fig. 4

Normalized ranked sensitivities computed for the log scaled parameters. Note that sensitivities decrease almost linearly and that all sensitivities are larger than the computational bound of 10⁻³.

Fig. 5

Subsets computed using the structured correlation method. Parameters noted in the tree are those appearing in pairwise correlations. Brackets below the tree show the actual parameters in each subset.

Fig. 6

Singular values associated with the sensitivity matrix.

Fig. 7

Subspace distance for subsets including 9 parameters. Note the gaps in the distances.

Fig. 8

Left panel shows simulated data and model result (an almost perfect match) and the right panel shows model results compared to the real data, both results are obtained with subset 1 identified using the structured correlation method.