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. 2021 Aug 18:386:114092.
doi: 10.1016/j.cma.2021.114092. eCollection 2021 Dec 1.

A computationally efficient physiologically comprehensive 3D-0D closed-loop model of the heart and circulation

Christoph M Augustin  1 , Matthias A F Gsell  1 , Elias Karabelas  1 , Erik Willemen  2 , Frits W Prinzen  2 , Joost Lumens  2 , Edward J Vigmond  3 , Gernot Plank  1   4
Affiliations

A computationally efficient physiologically comprehensive 3D-0D closed-loop model of the heart and circulation

Christoph M Augustin et al. Comput Methods Appl Mech Eng. .

Abstract

Computer models of cardiac electro-mechanics (EM) show promise as an effective means for the quantitative analysis of clinical data and, potentially, for predicting therapeutic responses. To realize such advanced applications methodological key challenges must be addressed. Enhanced computational efficiency and robustness is crucial to facilitate, within tractable time frames, model personalization, the simulation of prolonged observation periods under a broad range of conditions, and physiological completeness encompassing therapy-relevant mechanisms is needed to endow models with predictive capabilities beyond the mere replication of observations. Here, we introduce a universal feature-complete cardiac EM modeling framework that builds on a flexible method for coupling a 3D model of bi-ventricular EM to the physiologically comprehensive 0D CircAdapt model representing atrial mechanics and closed-loop circulation. A detailed mathematical description is given and efficiency, robustness, and accuracy of numerical scheme and solver implementation are evaluated. After parameterization and stabilization of the coupled 3D-0D model to a limit cycle under baseline conditions, the model's ability to replicate physiological behaviors is demonstrated, by simulating the transient response to alterations in loading conditions and contractility, as induced by experimental protocols used for assessing systolic and diastolic ventricular properties. Mechanistic completeness and computational efficiency of this novel model render advanced applications geared towards predicting acute outcomes of EM therapies feasible.

Keywords: Frank–Starling mechanism; Ventricular load; Ventricular pressure–volume relation.

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Conflict of interest statement

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Figures

Graphic abstract
Graphic abstract
Figure 1
Figure 1
Coarse (A) and fine (B) resolution meshes with domain labels and corresponding fiber fields. Note the difference in fiber angles due to spatial resolution.
Figure 2
Figure 2
Solution process of the lumped ODE model of the circulatory system. The CircAdapt model connects tubes (t), cavities (c), valves (v), and pulmonary and systemic periphery (py). In each timestep the ODE system is solved using a Runge-Kutta-Fehlberg method, see Appendix A.9, to update the ODE variables (in green), i.e., volumes of tubes (Vt) and cavities (Vc); sarcomere contractility (Cc) and sarcomere length (Lccont) for each of the cavities and the septum; flow over valves (qv); and septal midwall volume (VSepmid) and radius (y mid), see Fig. A.9c. In the following steps the updated variables are used to compute current pressures (pc, pt), cross sectional areas (Ac, At), and impedances (Zc, Zt) for tubes and cavities; fiber strain (Ecfib), fiber stiffness (κcfib), and fiber stress (σcfib) for the sarcomeres of each cavity and the septum; midwall curvature (Ccmid), midwall area (Acmid), and midwall tension (Tcmid) for each cavity and the septum; pericardial pressure p peri; and flow over the systemic and pulmonary periphery qpy.. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Source: Based on [44].
Figure 3
Figure 3
Schematic showing the coupling of the 0D ODE model, represented by the electrical equivalent circuit, to the 3D PDE model, represented by the FE mesh. In this case the ventricles (LV, RV) in the lumped model are replaced by 3D PDEs, while the atria (LA, RA) are modeled as lumped cavities in the CircAdapt model. Volume changes of the 3D cavities V˙LV, V˙RV are driven by flow q • of blood over valves and outlets computed by the 0D model. In turn, updated pressures p LV and p RV are used as an input to the lumped model in the next time step t n+1. The opening and closure of valves is only modeled in the lumped model and in the 3D model triangulated membranes are used to close the LV and RV cavities. Red colors indicate oxygenated and blue colors de-oxygenated blood. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Figure 4
Figure 4
EM activation of the coupled 3D–0D model is steered by an event-driven interconnected FSM that provides triggers for electrical activation of the 3D EM model and for mechanical activation of the 0D lumped atrial cavities. The sino-atrial node clock (SA) activates the RA at a prescribed cycle length, SACL, starting at time SAt0. The LA initiates contraction with a delay of AAdelay after the RA. The atrial entrance into the AV node activates at AVa which triggers, after the AVdelay elapsed, the ventricular exit of the AV node that is connected to the His bundle at AVv. Fascicles in the LV are activated then relative to the LV trigger to initiate electrical propagation in the EP model. Similarly, the RV is activated with an interventricular delay of VVd before (VVd < 0) or after VVd > 0 the LV.
Figure 5
Figure 5. Model parameterization under baseline conditions.
(A) The top panels show ventricular sinus activation sequence induced by three LV (fLV,a, fLV,s, fLV,p) and two RV fascicles (fmb). The bottom panels show the mechanical end-diastolic (red) and end-systolic (blue) configuration. Note the minor change in epicardial shape due to the pericardial boundary conditions. (B) Simulated pressure traces in LV (green) and RV (blue) are shown for the entire pacing protocol using a train of 20 beats. Envelopes (dotted traces) indicate that an approximate limit cycle was reached after 3 beats. (C) Left panels show time traces of pressure p, flow q and volume V in lumped 0D atrial cavities and PDE-based ventricular cavities for the last two beats of the limit cycle pacing protocol. Variables traverse the state space along limit cycle trajectories. Right panel shows pV loops in all four cavities. For PDE-based ventricular cavities EDPVR and ESPVR are indicated. Experimental data on peak LV pressure p^LV and stroke volume used for fitting are indicated (dashed lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Figure 6
Figure 6
Differences between the coarse (wireframe, red solid line) and higher resolution (solid, blue solid line) model are shown for (A) end-diastolic and (B) end-systolic configuration. (C) Dynamic behavior over the limit cycle protocol was comparable with minor difference in the stroke volume (SV) and peak pressure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Figure 7
Figure 7
Left ventricular pV loops showing the initial response (A–C) and 4 cycles (D–F) after applying a step change in loading conditions and contractility. (A) Altering afterload by increasing/decreasing the systemic vascular resistance, R sys pivots arterial elastance E a curve. Endsystolic elastance, E es and intercept V d characterizing the ESPVR was determined by linear regression of end-systolic data points V es and p es, marked by solid circles. (B) Increasing/decreasing preload shifts E a curve and increases/decreases stroke volume via the Starling mechanism, mediated by the length-dependence of the active stress model. Determination of ESPVR was consistent with afterload protocol. (C) Increasing/decreasing contractility increases/decreases stroke volume, LV peak pressure and p es. For each contractile state afterload was also perturbed to determine end-systolic elastance E es and V d.
Figure 8
Figure 8. Pump function graph (PFG) and Frank–Starling curve of the LV.
(A) pV loops in LV under baseline conditions (green) and varying afterload conditions, ranging between unloaded, E a ≈ 0, and isometric, E a ≈ ∞, conditions. Note that pV loops are plotted for the initial beat after altering afterload such that the end-diastolic volume is the same for all conditions. Thus, the system is not in a steady state and pV loops are therefore not closed. (B) PFG, plotting mean ventricular pressure (MVP), Eq. (20), against stroke volume (SV), constructed from afterload variations with constant preload and contractility (solid circles), with increased preload (solid squares) shifting the PFG up and left towards higher flow and pressure. For both cases contractility was also perturbed which pivots the PFG (empty circles and squares, respectively), leading to a steeper/flatter slope MVP/ΔSV for increased/decreased contractility. (C) Frank–Starling curve showing the relation between stroke volume and end-diastolic pressure, SV(p ed). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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