Regression Analysis for Constraining Free Parameters in Electrophysiological Models of Cardiac Cells

Abstract

A major challenge in computational biology is constraining free parameters in mathematical models. Adjusting a parameter to make a given model output more realistic sometimes has unexpected and undesirable effects on other model behaviors. Here, we extend a regression-based method for parameter sensitivity analysis and show that a straightforward procedure can uniquely define most ionic conductances in a well-known model of the human ventricular myocyte. The model's parameter sensitivity was analyzed by randomizing ionic conductances, running repeated simulations to measure physiological outputs, then collecting the randomized parameters and simulation results as “input” and “output” matrices, respectively. Multivariable regression derived a matrix whose elements indicate how changes in conductances influence model outputs. We show here that if the number of linearly-independent outputs equals the number of inputs, the regression matrix can be inverted. This is significant, because it implies that the inverted matrix can specify the ionic conductances that are required to generate a particular combination of model outputs. Applying this idea to the myocyte model tested, we found that most ionic conductances could be specified with precision (R ² > 0.77 for 12 out of 16 parameters). We also applied this method to a test case of changes in electrophysiology caused by heart failure and found that changes in most parameters could be well predicted. We complemented our findings using a Bayesian approach to demonstrate that model parameters cannot be specified using limited outputs, but they can be successfully constrained if multiple outputs are considered. Our results place on a solid mathematical footing the intuition-based procedure simultaneously matching a model's output to several data sets. More generally, this method shows promise as a tool to define model parameters, in electrophysiology and in other biological fields.

Author Summary

Mathematical models of biological processes generally contain many free parameters that are not known from experiments. Choosing values for these parameters, although an important step in the construction of realistic computational models, is frequently performed using an ad hoc approach that is a combination of intuition and trial and error. We have developed a novel method for constraining free parameters in mathematical models based on the techniques of linear algebra. We demonstrate this method's utility through simulations with a model of a human heart cell. The underlying premise is that if the model is only asked to recapitulate one or a few biological behaviors, the values of the parameters may be ambiguous; however, if the model must simultaneously match many features of experimental data, the free parameters can be determined uniquely. The results demonstrate that if computational models are to be realistic, they must be compared with several sets of data at the same time. This new method should serve as a valuable tool for investigators interested in developing realistic mathematical models of biological processes.