Uncertainty and variability in computational and mathematical models of cardiac physiology

Abstract

Key points: Mathematical and computational models of cardiac physiology have been an integral component of cardiac electrophysiology since its inception, and are collectively known as the Cardiac Physiome. We identify and classify the numerous sources of variability and uncertainty in model formulation, parameters and other inputs that arise from both natural variation in experimental data and lack of knowledge. The impact of uncertainty on the outputs of Cardiac Physiome models is not well understood, and this limits their utility as clinical tools. We argue that incorporating variability and uncertainty should be a high priority for the future of the Cardiac Physiome. We suggest investigating the adoption of approaches developed in other areas of science and engineering while recognising unique challenges for the Cardiac Physiome; it is likely that novel methods will be necessary that require engagement with the mathematics and statistics community.

Abstract: The Cardiac Physiome effort is one of the most mature and successful applications of mathematical and computational modelling for describing and advancing the understanding of physiology. After five decades of development, physiological cardiac models are poised to realise the promise of translational research via clinical applications such as drug development and patient-specific approaches as well as ablation, cardiac resynchronisation and contractility modulation therapies. For models to be included as a vital component of the decision process in safety-critical applications, rigorous assessment of model credibility will be required. This White Paper describes one aspect of this process by identifying and classifying sources of variability and uncertainty in models as well as their implications for the application and development of cardiac models. We stress the need to understand and quantify the sources of variability and uncertainty in model inputs, and the impact of model structure and complexity and their consequences for predictive model outputs. We propose that the future of the Cardiac Physiome should include a probabilistic approach to quantify the relationship of variability and uncertainty of model inputs and outputs.

Figure 1

Illustrative example showing how model inputs (I1, I2) and outputs (O) can be characterised as probability distributions rather than fixed values

Figure 2

Diagram showing how different sources of uncertainty combine to produce output uncertainty (continuous arrows), and how structural uncertainty is important for model calibration (dashed arrows)

Figure 3. Uncertainty propagation in an action potential model

A, action potentials produced by the ten Tusscher–Panfilov 2006 model when paced at a 1000 ms cycle length for 20 beats in its default configuration (blue), and with G _Ks altered by ±20% (red). B, a series of 20 action potentials in which the values of G _Ks are drawn from a uniform distribution with range ±20%. C, a series of 20 action potentials in which G _Ks is drawn from a normal distribution with a coefficient of variation (standard deviation divided by mean) of 10%.

Figure 4. Simulations of fibrillation in an ellipsoid geometry representing the human LV, with membrane kinetics described by a phenomenological model (Bueno‐Orovio et al. 2008 ) set to steep APD restitution (Clayton, 2013 )

Three snapshots of a baseline simulation are shown (left column), along with four further simulations (P1–P4) in which the initial conditions of a single re‐entrant wave with a transmural filament are perturbed by adding a random voltage drawn from a uniform distribution in the range ±5 mV to the initial voltage at each grid point (Qu et al. 2000). The graph on the right shows the nonlinear growth of the average absolute difference at each grid point between the voltage in the baseline simulation and each of the perturbations.

Figure 5. A simple Gaussian process emulator, which relates a single output (APD) to a single input (G_K) for the Luo–Rudy 1991 model (Luo & Rudy, 1991 )

Circles denote training data, blue line is mean of emulator, green lines are two standard deviations, red lines show distribution of output for a given input distribution. See text for further details.