Uncertainty and variability in models of the cardiac action potential: Can we build trustworthy models?

Abstract

Cardiac electrophysiology models have been developed for over 50years, and now include detailed descriptions of individual ion currents and sub-cellular calcium handling. It is commonly accepted that there are many uncertainties in these systems, with quantities such as ion channel kinetics or expression levels being difficult to measure or variable between samples. Until recently, the original approach of describing model parameters using single values has been retained, and consequently the majority of mathematical models in use today provide point predictions, with no associated uncertainty. In recent years, statistical techniques have been developed and applied in many scientific areas to capture uncertainties in the quantities that determine model behaviour, and to provide a distribution of predictions which accounts for this uncertainty. In this paper we discuss this concept, which is termed uncertainty quantification, and consider how it might be applied to cardiac electrophysiology models. We present two case studies in which probability distributions, instead of individual numbers, are inferred from data to describe quantities such as maximal current densities. Then we show how these probabilistic representations of model parameters enable probabilities to be placed on predicted behaviours. We demonstrate how changes in these probability distributions across data sets offer insight into which currents cause beat-to-beat variability in canine APs. We conclude with a discussion of the challenges that this approach entails, and how it provides opportunities to improve our understanding of electrophysiology.

Keywords: Cardiac electrophysiology; Mathematical model; Probability; Uncertainty quantification.

Fig. 1

Pipeline of the UQ process, applied to variability in steady-state I_Na inactivation (image adapted from [38]). Using experimental data on steady-state I_Na inactivation from canine myocytes (sub-figure (a), individual cells in grey, averaged data in black, note how the slope of the average does not match the slope of any individual), uncertainty due to population variability was characterised by fitting a two-parameter sigmoidal curve using the statistical method nonlinear mixed effects to estimate the variability in the two parameters. Sub-figure (b) illustrates the mean (red star) and variability (red ellipse: 95%; grey ellipses: 80% and 99%) of the two parameters (blue stars represent parameters for individual cells). The mean corresponds to an ‘average’ cell, not the averaged data. This inactivation sub-model was then embedded in the Fox et al. canine AP model, and the parameter uncertainty was propagated through the model to obtain a probability distribution for upstroke velocity (subfigure (c)). However, the model did not repolarise for a region of parameter space that overlapped with the population variability (subfigure (d)).

Fig. 2

Top: synthetic data example of minimisation and MCMC for the Luo & Rudy model : minimisation of the negative log-likelihood to find starting point for MCMC followed by log-likelihoods obtained during MCMC. The MCMC begins at the vertical red line, note the log scale on the x-axis — minimisation is a much smaller proportion of the iterations than MCMC. Points A, B and C are at iterations 8, 100 and 2000, respectively. Inset is a zoomed in view of the highlighted box, showing the MCMC exploring the likelihood between iterations 2000 and 3000. Panels A–C: the blue traces are the APs for the proposed parameters at iterations marked as A, B and C. Red traces are the same synthetic data in all 3 plots. Panels D–F: three experimentally recorded canine APs are shown in red in panels D, E and F. The green trace is the Davies et al. 1 Hz steady pacing AP , and the blue trace is the same model with conductances inferred to fit the experimental recordings.

Fig. 3

Marginal and 2-d normalised histograms representing marginal and pairwise probability distributions of each parameter, using the Beeler & Reuter model model and single AP protocol. All parameters have been successfully inferred from the data (see main text). The vertical red (outside) lines give the 95% credible interval. The vertical green (inside) line is the parameter value used to generate the synthetic data.

Fig. 4

Subset of the histograms and pairwise probability distributions of each parameter using the ten Tusscher et al. (2004) model and single AP protocol for synthetic data. Not all parameters have been successfully inferred from the data, e.g. G_bCa , G_bNa and P_NaK. P_NaK has the shape of a typically converged distribution, but fails our test of success by being too wide. The vertical red (outside) lines give the 95% credible interval. The green (inside) lines are the parameter values used to generate the synthetic data.

Fig. 5

Posterior probability distribution estimates of current densities, and the noise parameter σ, inferred from a series of canine cardiomyocyte APs. While some currents are relatively consistent, e.g. G_K1, others vary by a wide range, e.g. G_Kr, G_Ks. This figure should be considered in the light of the results in Table 1 (and Supplementary Material Table B1), which shows those currents for which we expect to be able to recover conductances using a single 1 Hz AP. The colours transition from semi-transparent blue to semi-transparent green through time.

Fig. 6

TP06 design data for 50 model runs (coloured circles), and test data obtained from 10 additional model runs (grey circles).

Fig. 7

(a, b) Mean effects plots for (a) Max. dV_m/dt and (b) APD₉₀ emulators, showing how varying each input in the range from 0 to 1 influences the expected value of the emulator output. (c, d) Cumulative effect of changing the variance of inputs on emulator outputs. (c) Distributions of Max. dV_m/dt when G_Na was reduced from 0.04 (red), to 0.02 (blue), 0.01 (green), and 0.005 (yellow) while the variance of all other inputs was maintained at 0.04 normalised units. (d) Distributions of APD₉₀ when the variance of all inputs was initially set to 0.001 normalised units, and then the variance of G_Kr (red), G_Ks (blue), G_CaL (green), and then τ_f multiplier (yellow) were set to 0.04 normalised units.

Fig. 8

Main effect sensitivity indices for each TP06 emulator.