Exploiting mathematical models to illuminate electrophysiological variability between individuals

Abstract

Across individuals within a population, several levels of variability are observed, from the differential expression of ion channels at the molecular level, to the various action potential morphologies observed at the cellular level, to divergent responses to drugs at the organismal level. However, the limited ability of experiments to probe complex interactions between components has hitherto hindered our understanding of the factors that cause a range of behaviours within a population. Variability is a challenging issue that is encountered in all physiological disciplines, but recent work suggests that novel methods for analysing mathematical models can assist in illuminating its causes. In this review, we discuss mathematical modelling studies in cardiac electrophysiology and neuroscience that have enhanced our understanding of variability in a number of key areas. Specifically, we discuss parameter sensitivity analysis techniques that may be applied to generate quantitative predictions based on considering behaviours within a population of models, thereby providing novel insight into variability. Our discussion focuses on four issues that have benefited from the utilization of these methods: (1) the comparison of different electrophysiological models of cardiac myocytes, (2) the determination of the individual contributions of different molecular changes in complex disease phenotypes, (3) the identification of the factors responsible for the variable response to drugs, and (4) the constraining of free parameters in electrophysiological models of heart cells. Together, the studies that we discuss suggest that rigorous analyses of mathematical models can generate quantitative predictions regarding how molecular-level variations contribute to functional differences between experimental samples. These strategies may be applicable not just in cardiac electrophysiology, but in a wide range of disciplines.

Figure 1. A comparison of two paradigms in modelling

A, the traditional approach represents a model as occupying a single point in parameter space (left) and a single point in measurement or output space (right). B, a newer approach towards modelling takes variability into account. Under this paradigm, it is more useful to consider a population of models, each with slightly different parameters. This population occupies regions in both parameter space and output space. Analysis is required, however, to understand how changes in parameters translate to changes in outputs.

Figure 2. Three-dimensional plot showing the transition from control to an altered condition in response to changes in parameter values

A particular cell, with defined values of 3 physiological outputs, resides in a specific location in output space. The three outputs in this example are action potential duration (APD), Ca²⁺ transient amplitude (Δ[Ca²⁺]_i) and action potential upstroke velocity (dV/dt_max). By convention, the control cell, shown in blue, resides at the origin. When values of two parameters (here G_Na and G_Ca) change, the final location of the cell, shown in red, can be calculated as the sum of two vectors, each of which is the product of a parameter change and the corresponding vector of parameter sensitivities. These are shown to the right of the 3D plot. The first step, due to an increase in G_Na, causes movement only along the upstroke velocity axis because this is the only non-zero parameter sensitivity. The second step, a decrease in G_Ca, leads to a decrease in APD and a decrease in Δ[Ca²⁺]_i, but virtually no change in dV/dt_max.

Figure 3. Multivariable regression applied to predict effects of changes occurring during heart failure

Eight parameter changes corresponding to heart failure, as previously simulated (Shannon et al. 2005), are implemented in a mathematical model of the human ventricular myocyte (Grandi et al. 2010). A, the heart failure simulation predicted an increase in APD. B, the heart failure simulation predicted a decrease in Δ[Ca²⁺]_i. C, the matrix multiplication approach after performing multivariable regression allows us to correctly predict these changes in APD and Δ[Ca²⁺]_i by summing up the independent contributions of each of the eight parameter changes. This approach also reveals which parameter changes are most responsible for the observed changes in phenotype. For example, NCX and G_K1 both contribute greatly to APD prolongation while NCX and K_leak are predominantly responsible for the decrease in calcium transient amplitude. D, compensatory changes occur to maintain APD at control value when NCX and G_K1 are both reduced to 0.5 and 0.7 times their control values, respectively. For these simulations, the particular parameter changes implemented, represented graphically in C, are as follows: decrease in maximal slow transient outward K⁺ conductance (G_toslow), decrease in maximal fast transient outward K⁺ conductance (G_tofast), decrease in maximal slow delayed rectifier K⁺ current conductance (G_Ks), decrease in maximal inward rectifier K⁺ conductance (G_K1), increase in maximal Na⁺–Ca²⁺ exchange current (NCX), increase in SR Ca²⁺ release (K_s), increase in passive SR Ca²⁺ leak (K_leak), and decrease in maximal rate of SR Ca²⁺ uptake (V_maxSRCaP).

Figure 4. Analogous findings in the fields of neuroscience and cardiac electrophysiology

Virtually indistinguishable electrical activity can result from very different sets of conductances. A, voltage traces from three model neurons look essentially identical. The underlying conductances shown at right, however, are quite different (Golowasch et al. 2002, reprinted with permission) B, similar to the behaviour observed in neuronal models, drastically different conductances in two model variants can produce identical action potentials in a mathematical model of the human ventricular myocyte (Sarkar & Sobie, 2010, reprinted with permission).