Sensitivity analysis of a strongly-coupled human-based electromechanical cardiac model: Effect of mechanical parameters on physiologically relevant biomarkers

Abstract

The human heart beats as a result of multiscale nonlinear dynamics coupling subcellular to whole organ processes, achieving electrophysiologically-driven mechanical contraction. Computational cardiac modelling and simulation have achieved a great degree of maturity, both in terms of mathematical models of underlying biophysical processes and the development of simulation software. In this study, we present the detailed description of a human-based physiologically-based, and fully-coupled ventricular electromechanical modelling and simulation framework, and a sensitivity analysis focused on its mechanical properties. The biophysical detail of the model, from ionic to whole-organ, is crucial to enable future simulations of disease and drug action. Key novelties include the coupling of state-of-the-art human-based electrophysiology membrane kinetics, excitation-contraction and active contraction models, and the incorporation of a pre-stress model to allow for pre-stressing and pre-loading the ventricles in a dynamical regime. Through high performance computing simulations, we demonstrate that 50% to 200% $-$ 1000% variations in key parameters result in changes in clinically-relevant mechanical biomarkers ranging from diseased to healthy values in clinical studies. Furthermore mechanical biomarkers are primarily affected by only one or two parameters. Specifically, ejection fraction is dominated by the scaling parameter of the active tension model and its scaling parameter in the normal direction ( $k_{\text{ort}2}$ ); the end systolic pressure is dominated by the pressure at which the ejection phase is triggered ( $P_{\text{ej}}$ ) and the compliance of the Windkessel fluid model ( $C$ ); and the longitudinal fractional shortening is dominated by the fibre angle ( $\varphi$ ) and $k_{\text{ort}2}$ . The wall thickening does not seem to be clearly dominated by any of the considered input parameters. In summary, this study presents in detail the description and implementation of a human-based coupled electromechanical modelling and simulation framework, and a high performance computing study on the sensitivity of mechanical biomarkers to key model parameters. The tools and knowledge generated enable future investigations into disease and drug action on human ventricles.

Keywords: Cardiac electromechanics; Finite element method; High-performance computing; Multiscale simulations; Sensitivity analysis.

Fig. 2.1

Diagram of the human ventricular electromechanical modelling and simulation framework, which includes: (blue) nonlinear solid mechanics, (green) electrical propagation, (yellow) ventricular cell electrophysiology, and (red) active-tension plus excitation-contraction. The arrows represent the different coupling mechanisms and their corresponding directions, with the quantities written in red being exchanged between the different components of the model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2.2

Left ventricular geometry, including local fibre architecture and domain boundaries. The centreline of the ventricular cavity is aligned with the $x_3$-axis.

Fig. 3.1

(a) Sequence of snapshots of contraction throughout the whole heartbeat, the colour code indicates transmembrane potential (red, fully activated and depolarised, and blue, polarised); (b) pressure–volume loop throughout the whole heartbeat, A–E correspond to the snapshots in (a); (c) a sketch depicting wall thickness and apico-basal distance (which are the absolute counterparts of wall thickening (WT) and longitudinal fractional shortening (LFS)). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3.2

Mechanical biomarkers versus one-at-a-time variations of the considered parameters. (a) Ejection fraction versus $\{k_{\text{epi}},P_0,P_{\text{ej}},C,K\}$. (b) Ejection fraction versus $\{R,k_{\text{ort}2},T_{\text{ref}},\varphi ,a_{\text{s}}\}$. (c) End-systolic pressure versus $\{k_{\text{epi}},P_0,P_{\text{ej}},C,K\}$. (d) End-systolic pressure versus $\{R,k_{\text{ort}2},T_{\text{ref}},\varphi ,a_{\text{s}}\}$. (e) Longitudinal fractional shortening versus $\{k_{\text{epi}},P_0,P_{\text{ej}},C,K\}$. (f) Longitudinal fractional shortening versus $\{R,k_{\text{ort}2},T_{\text{ref}},\varphi ,a_{\text{s}}\}$. (g) Wall-thickening versus $\{k_{\text{epi}},P_0,P_{\text{ej}},C,K\}$. (h) Wall-thickening versus $\{R,k_{\text{ort}2},T_{\text{ref}},\varphi ,a_{\text{s}}\}$. Areas shaded in green represent the healthy ranges of the considered mechanical biomarkers. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3.3

(Left) Mechanical deformation of the left ventricular geometry from diastole (transparent grey) to systole (red) for two cases with different mechanical parameter values. Left panel corresponds to low pericardial stiffness ($k_{\text{epi}}$) or low fibre angle ($\varphi$), whereas right panel results from high $k_{\text{epi}}$ or high $\varphi$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3.4

Histograms of simulated mechanical biomarkers compared to clinical values. The green shaded area indicates clinical ranges for healthy patients (Table 2.3). (a) Ejection fraction. The grey shaded area corresponds to low ejection fraction ($<$35% is the threshold for implantable cardioverter-defibrillator indicated by clinical guidelines [68]). The dashed red line indicates the clinical mean value for heart failure patients. . (b) End-systolic pressure. The dashed red line indicates the clinical mean value for dilated cardiomyopathy patients . (c) Longitudinal fractional shortening. The dashed red lines indicate the clinical mean values for patients with coronary artery disease , left, and high blood pressure with supranormal ejection fraction , right. (d) Wall thickening. The dashed red line corresponds to the clinical mean value for patients with ischaemic heart disease with dyskinetic segments . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 3.5

Correlation plot showing the partial rank correlation coefficients between each of the varied parameters and each of the simulated mechanical biomarkers. White grids have almost zero correlation coefficients and thus an associated $p$-value lower than the threshold for statistical significance ($p<0.05$). “Plus” or “minus” symbols only appear in the statistically significant values of the correlation coefficients and indicate a positive or a negative correlation, respectively.

Fig. 3.6

Scatter-plots of the mechanical biomarkers with respect to their most influential model parameters. (a) Ejection fraction versus $T_{\text{ref}}$. (b) Ejection fraction versus $k_{\text{ort}2}$. (c) End-systolic pressure versus $P_{\text{ej}}$. (d) End-systolic pressure versus $k_{\text{ort}2}$. (e) Longitudinal fractional shortening versus $\varphi$. (f) Longitudinal fractional shortening versus $k_{\text{ort}2}$. Black circles denote “healthy” cases (ejection fraction higher than 40%), whereas grey circles denotes “diseased” cases (ejection fraction lower than 40%). Coloured circles denote cases that maximise a mechanical biomarker (red for ejection fraction, green for end-systolic pressure and blue for longitudinal fractional shortening). On the other hand, coloured crosses denote cases that minimise a mechanical biomarker. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)