Bayesian Sensitivity Analysis of a Cardiac Cell Model Using a Gaussian Process Emulator

Abstract

Models of electrical activity in cardiac cells have become important research tools as they can provide a quantitative description of detailed and integrative physiology. However, cardiac cell models have many parameters, and how uncertainties in these parameters affect the model output is difficult to assess without undertaking large numbers of model runs. In this study we show that a surrogate statistical model of a cardiac cell model (the Luo-Rudy 1991 model) can be built using Gaussian process (GP) emulators. Using this approach we examined how eight outputs describing the action potential shape and action potential duration restitution depend on six inputs, which we selected to be the maximum conductances in the Luo-Rudy 1991 model. We found that the GP emulators could be fitted to a small number of model runs, and behaved as would be expected based on the underlying physiology that the model represents. We have shown that an emulator approach is a powerful tool for uncertainty and sensitivity analysis in cardiac cell models.

Fig 1. Outputs produced by the LR91 model.

(a) Action potential biomarkers used as model outputs to characterise the model. (b) Action potential time series from 200 runs of the LR1991 model used as design data for the GP emulator.

Fig 2. Design data and test data obtained from 220 runs of the LR1991 model for eight outputs and six inputs.

Each plot shows combinations of inputs and outputs, with 200 coloured points indicating design data used to fit the emulators, and 20 grey points showing test data used to validate the emulator.

Fig 3. Validation of APD₉₀ emulator output against test data output.

This plot shows the difference in the mean APD₉₀ predicted by the emulator and APD₉₀ obtained from the simulator for each of the 20 test data. The difference is calibrated as the number of standard deviations, and the red lines indicate ±2 standard deviations.

Fig 4. Variance in APD₉₀ emulator.

(a) Distributions of APD₉₀ resulting from normal distributions of G_K when all other maximum conductances were effectively held constant by assigning a mean of 0.5 and a very small variance of 0.0001 in normalised units. G_K was assigned a mean value of 0.282 mS cm^-2 and variance 0.0014 (blue) 0.0028 (red), 0.0071 (green), 0.0141 (yellow) mS cm^-2. (b) Distributions of APD₉₀ obtained from four Monte Carlo analyses, each with 2000 simulator runs, and with G_K drawn from distributions with mean value of 0.282 mS cm^-2 and variance 0.0014 (blue) 0.0028 (red), 0.0071 (green), 0.0141 (yellow) mS cm^-2.

Fig 5. Mean effects in APD₉₀ emulator.

Mean effect of each of the inputs on APD₉₀ as each input is varied whilst the others are held at their mean value.

Fig 6. Sensitivity indices in APD₉₀ emulator.

Sensitivity index of each of the inputs on APD₉₀, showing the proportion of total variance that can be attributed to variance in each input.

Fig 7. Mean effects in each emulator.

Mean effect of each of the inputs on each output as each input is varied whilst the others are held at their mean value.

Fig 8. Variance based main effect indices for each emulator.

Main effect index of each emulator (rows) to each input (columns), describing the proportion of the output variance that can be accounted for by variance on the input.

Fig 9. Sensitivity indices calculated with PLS technique.

Sensitivity index of each emulator (rows) to each input (columns).