# Electro-Mechanical Whole-Heart Digital Twins: A Fully Coupled Multi-Physics Approach

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Four-Chamber Heart Model

#### 2.2. Cardiac Elasto-Mechanics

#### 2.2.1. Contact Boundary Conditions

#### 2.2.2. Closed-Loop Circulatory Model

#### 2.3. Cardiac Electrical Activity

#### 2.4. Electro-Mechanical Coupling Mechanisms

#### 2.4.1. Cellular Electro-Mechanical Model

#### 2.4.2. Mechano-Electric Feedback

#### 2.5. Electro-Mechanical Coupling Algorithm

## 3. Patient-Specific Simulation and Evaluation

#### 3.1. Personalizing Electro-Mechanical Whole Heart Models: Building Digital Twins

#### 3.1.1. Cardiac Anatomy

#### 3.1.2. Fiber Orientation

#### 3.1.3. Passive Stress

#### 3.1.4. Active Stress

#### 3.1.5. Electrophysiology

#### 3.2. Experimental Setup

#### 3.2.1. Parameterization

#### 3.2.2. Initialization

#### 3.2.3. Evaluation

## 4. Results

#### 4.1. Cellular Electro-Mechanical Model

#### 4.2. Electro-Mechanical Whole-Heart Simulation

## 5. Discussion

#### 5.1. Bidirectional Coupling between the Mechanical and Electrophysiological Systems

#### 5.2. Circulatory System

#### 5.3. Numerical Considerations

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) triangulated surfaces used for the boundary conditions: ${\mathsf{\Gamma}}_{\mathrm{N}}={\mathsf{\Gamma}}_{\mathrm{LV}}\cup {\mathsf{\Gamma}}_{\mathrm{RV}}\cup {\mathsf{\Gamma}}_{\mathrm{LA}}\cup {\mathsf{\Gamma}}_{\mathrm{RA}}$ for the pressure in the left and right ventricle and atrium; ${\mathsf{\Gamma}}_{\mathrm{D}}$ for the dirichlet boundary; ${\mathsf{\Gamma}}_{\mathrm{P}}$ for the pericardium. (

**b**) clipped reference domain ${\mathsf{\Omega}}_{\mathrm{M}}$ of the patient specific heart.

**Figure 2.**Schematic of the circulatory system model with the pressure values of $\mathbf{p}$ and $\mathbf{z}$, resistances R, fixed compliances C and volumes of variable compliances ${v}_{\mathrm{C}}$ with $\mathrm{C}\in \left\{\mathrm{LV},\phantom{\rule{4.pt}{0ex}}\mathrm{RV},\phantom{\rule{4.pt}{0ex}}\mathrm{LA},\phantom{\rule{4.pt}{0ex}}\mathrm{RA}\right\}$.

**Figure 3.**Staggered algorithm for the fully coupled problem at time steps ${t}_{n+1}=(n+1)\Delta {t}_{\mathrm{M}}$ evaluating the solutions at ${t}_{n}=n\Delta {t}_{\mathrm{M}}$ with different time step sizes $\Delta {t}_{\mathrm{M}}=0.001$ s, $\Delta {t}_{\mathrm{EP}}=0.00001$ s and $\Delta {t}_{{\scriptstyle \mathbf{p}}}=0.0001$ s for $n>5$ and $\mathrm{tol}={10}^{-7}$ mL.

**Figure 5.**Anterior (left) and posterior (right) view of the atria with labels for fast conducting and slow conducting materials, as well as the scar. Transparant volumes represent the atrial bulk tissue.

**Figure 7.**Action potential (left), CaT (center), and active tension (right) of the optimized (solid line) and the original (dashed line) atrial model with reference experimental values from literature [129].

**TPCaT:**time to peak of calcium transient;

**TPT:**time to peak tension;

**RT50/90/95:**relaxation times to 50/90/95% decay from peak calcium/tension. Only the last cycle is visualized.

**Figure 8.**Action potential (left), CaT (center), and active tension (right) of the optimized (solid line) and the original (dashed line) ventricular model with reference experimental values from literature [125].

**TPCaT:**time to peak of calcium transient;

**TPT:**time to peak tension;

**RT50/90/95:**relaxation times to 50/90/95% decay from peak calcium/tension. Only the last cycle is visualized.

**Figure 9.**Snapshots of the deformation of the heart during the final cycle. Visualized is the stretch in fiber direction ${\gamma}_{{\scriptstyle \mathbf{f}}}$ in a clipped long axis four chamber view. (

**a**) at rest; (

**b**) end-diastole; (

**c**) end-systole; (

**d**) end of isovolumetric relaxation.

**Figure 10.**Results of the circulatory system for the reference simulation (solid line) and the simulation with an ablation scar in the left atrium (dashed line) after reaching a stable limit cycle. First column from top to bottom: (1) Wiggers diagram showing pressure with respect to time; (2) cavity volume with respect to time; (3) flow through mitral valve (LAV), tricuspid valve (RAV), systemic arterial flow (SysArt), and pulmonary arterial flow (PulArt). The second and third column show the phase diagrams of the pressure-volume relationship for the left atrium (LA), left ventricle (LV), right atrium (RA), and right ventricle (RV).

**Figure 11.**Left ventricular (LV) cavity volume derived from imaging data (black) and numerical simulation (red). LV volume was normalized to the maximally measured volumes.

**Figure 12.**Atrioventricular plane displacement from imaging data (dashed line) and numerical simulation (solid line).

**Figure 13.**Atrial local activation times from the reference simulation (left) and the simulation including an ablation scar in the left atrium (right).

Parameter | Value | Unit | Description |
---|---|---|---|

$({\sigma}_{\mathrm{f}},{\sigma}_{\mathrm{s}},{\sigma}_{\mathrm{n}})$ | $(0.47,0.27,0.15)$ | S/m | conductivities in ventricular bulk tissue |

$({\sigma}_{\mathrm{f}},{\sigma}_{\mathrm{s}},{\sigma}_{\mathrm{n}})$ | $(3.25,2.75,2.25)$ | S/m | conductivities in ventricular fast conducting layer |

$({\sigma}_{\mathrm{f}},{\sigma}_{\mathrm{s}},{\sigma}_{\mathrm{n}})$ | $(0.99,0.26,0.26)$ | S/m | conductivities in atrial bulk tissue |

$({\sigma}_{\mathrm{f}},{\sigma}_{\mathrm{s}},{\sigma}_{\mathrm{n}})$ | $(2.35,0.99,0.99)$ | S/m | conductivities in atrial fast conducting regions |

$({\sigma}_{\mathrm{f}},{\sigma}_{\mathrm{s}},{\sigma}_{\mathrm{n}})$ | $(0.26,0.26,0.26)$ | S/m | conductivities in atrial slow conducting regions |

$({\sigma}_{\mathrm{f}},{\sigma}_{\mathrm{s}},{\sigma}_{\mathrm{n}})$ | $({10}^{-12},{10}^{-12},{10}^{-12})$ | S/m | conductivities in scar tissue |

$\beta $ | 140,000 | 1/m | membrane surface-to-volume ratio |

${C}_{\mathrm{m}}$ | $0.01$ | F/m${}^{2}$ | membrane capacitance |

AV-delay | $0.160$ | s | atrio-ventricular conduction delay |

BCL | 1 | s | basic cycle length (=$1/\mathrm{heartrate}$) |

**Table 2.**Sites of earliest activation in terms of the coordinate system Cobiveco [42]: a apicobasal; m transmural; r rotational; v transventricular.

Root Point | Extent | ||
---|---|---|---|

$\mathbf{x}$ | ${{\textstyle \mathbf{x}}}_{\mathbf{root}}=\{\mathit{a},\mathit{m},\mathit{r},\mathit{v}\}$ | ${\mathbf{\delta}}_{\mathbf{m}}$ | ${\mathbf{\delta}}_{\mathbf{rad}}$ |

${{\textstyle \mathbf{x}}}_{\mathrm{LV},\mathrm{mps}}$ | $\{0.65,1,0.99,0\}$ | $0.05$ | 3 mm |

${{\textstyle \mathbf{x}}}_{\mathrm{LV},\mathrm{mpi}}$ | $\{0.55,1,0.175,0\}$ | $0.05$ | 3 mm |

${{\textstyle \mathbf{x}}}_{\mathrm{LV},\mathrm{bap}}$ | $\{0.9,1,0.57,0\}$ | $0.05$ | 3 mm |

${{\textstyle \mathbf{x}}}_{\mathrm{RV},\mathrm{s}}$ | $\{0.4,1,0.85,1\}$ | $0.05$ | 3 mm |

${{\textstyle \mathbf{x}}}_{\mathrm{RV},\mathrm{fw}}$ | $\{0.45,1,0.15,1\}$ | $0.1$ | 3 mm |

**Table 3.**Overview of passive mechanical parameters in ${W}_{\mathrm{G}},{W}_{\mathrm{NH}}$ for the whole heart model.

Parameters | |||||||
---|---|---|---|---|---|---|---|

Domain | Model | $\mathit{\mu}$ (Pa) | ${\mathit{b}}_{{\scriptstyle \mathit{f}}}$ | ${\mathit{b}}_{{\scriptstyle \mathit{s}}}$ | ${\mathit{b}}_{{\scriptstyle \mathit{f}}{\scriptstyle \mathit{s}}}$ | $\mathit{\kappa}$ (Pa) | ${\mathit{\rho}}_{0}$(kg/m${}^{2}$) |

${\mathsf{\Omega}}_{\mathrm{V}}$ | Guccione | $325.56$ | $11.01$ | $4.4$ | $7.71$ | ${10}^{6}$ | 1082 |

${\mathsf{\Omega}}_{\mathrm{A}}$ | Guccione | $325.56$ | $11.01$ | $4.4$ | $7.71$ | ${10}^{6}$ | 1082 |

${\mathsf{\Omega}}_{\mathrm{Valves}}$ | Neo-Hooke | ${10}^{5}$ | - | - | - | ${10}^{3}$ | 1082 |

${\mathsf{\Omega}}_{\mathrm{AdiposeTissue}}$ | Neo-Hooke | 3725 | - | - | - | ${10}^{3}$ | 1082 |

${\mathsf{\Omega}}_{\mathrm{MajorVessels}}$ | Neo-Hooke | ${10}^{4}$ | - | - | - | ${10}^{3}$ | 1082 |

${\mathsf{\Omega}}_{\mathrm{Pericardium}}$ | Neo-Hooke | ${10}^{4}$ | - | - | - | ${10}^{3}$ | 1082 |

**Table 4.**Overview of circulatory system parameters in Section 2.2.2 for the whole heart model.

Parameter | Value | Unit | Description | Ref. |
---|---|---|---|---|

${R}_{\mathrm{SysArtValve}}$ | 0.006 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | aortic valve resistance | [110,111] |

${R}_{\mathrm{SysArt}}$ | 0.03 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | systemic arterial resistance | [112,113] |

${C}_{\mathrm{SysArt}}$ | 3.0 | $\mathrm{mL}\xb7{\mathrm{mmHg}}^{-1}$ | systemic arterial compliance | [112,113,114] |

${V}_{\mathrm{SysArtUnstr}}$ | 800.0 | mL | unstressed systemic arterial volume | [110] |

${R}_{\mathrm{SysPer}}$ | 0.6 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | systemic peripheral resistance | [112,113] |

${R}_{\mathrm{SysVen}}$ | 0.03 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | systemic venous resistance | [115,116] |

${C}_{\mathrm{SysVen}}$ | 150.0 | $\mathrm{mL}\xb7{\mathrm{mmHg}}^{-1}$ | systemic venous compliance | [110,111,116] |

${V}_{\mathrm{SysVenUnstr}}$ | 2850.0 | mL | unstressed systemic venous resistance | [110,115] |

${R}_{\mathrm{RavValve}}$ | 0.003 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | tricuspid valve resistance | [20,111] |

${R}_{\mathrm{PulArtValve}}$ | 0.003 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | pulmonary valve resistance | [110] |

${R}_{\mathrm{PulArt}}$ | 0.02 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | pulmonary arterial resistance | [117,118] |

${C}_{\mathrm{PulArt}}$ | 10.0 | $\mathrm{mL}\xb7{\mathrm{mmHg}}^{-1}$ | pulmonary arterial compliance | [117,119] |

${V}_{\mathrm{PulArtUnstr}}$ | 150.0 | mL | unstressed pulmonary arterial volume | [110] |

${R}_{\mathrm{PulPer}}$ | 0.07 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | pulmonary peripheral resistance | [120,121] |

${R}_{\mathrm{PulVen}}$ | 0.03 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | pulmonary venous resistance | [20] |

${C}_{\mathrm{PulVen}}$ | 15.0 | $\mathrm{mL}\xb7{\mathrm{mmHg}}^{-1}$ | pulmonary venous compliance | [119] |

${V}_{\mathrm{PulVenUnstr}}$ | 200.0 | mL | unstressed pulmonary venous volume | [110] |

${R}_{\mathrm{LavValve}}$ | 0.003 | $\mathrm{mmHg}\xb7\mathrm{s}\xb7{\mathrm{mL}}^{-1}$ | mitral valve resistance | [20] |

Parameter | Value | Unit | Description |
---|---|---|---|

${V}_{\mathrm{tot}}$ | 5700.0 | mL | total volume |

${V}_{\mathrm{SysArt}}$ | 981.1396 | mL | systemic arterial volume |

${V}_{\mathrm{PulArt}}$ | 303.7683 | mL | pulmonary arterial volume |

${V}_{\mathrm{PulVen}}$ | 349.6759 | mL | pulmonary venous volume |

${p}_{\mathrm{LV}}$ | 8.0246 | mmHg | left ventricular pressure |

${p}_{\mathrm{LA}}$ | 8.2061 | mmHg | left atrial pressure |

${p}_{\mathrm{RV}}$ | 5.8073 | mmHg | right ventricular pressure |

${p}_{\mathrm{RA}}$ | 5.8071 | mmHg | right atrial pressure |

**Table 6.**Literature values for time to peak of calcium transient (TPCaT) and active tension development (TPT) as well as relaxation time to 50%, 90% and 95% respectively (RT50, RT90, RT95) from human tissue preparations. The list of ventricular values were originally compiled in [36].

Calcium Transient | Active Tension | ||||||
---|---|---|---|---|---|---|---|

Tissue | TPCaT (ms) | RT50 (ms) | RT90 (ms) | TPT (ms) | RT50 (ms) | RT95 (ms) | Ref. |

Ventricle | $47.8\pm 10.0$ | $151.1\pm 89.2$ | $315.6\pm 161.2$ | - | - | - | [124] |

Ventricle | - | - | - | $165\pm 7$ | $116\pm 6$ | $334\pm 43$ | [125] |

Ventricle | - | - | - | $157\pm 10$ | $117\pm 8$ | $477\pm 31$ | [126] |

Ventricle | - | - | - | $235.0\pm 13.4$ | $153\pm 71$ | $309\pm 13.7$ | [127] |

Ventricle | - | - | - | $151.0\pm 6.1$ | $98.0\pm 7.7$ | $173.0\pm 10.7$ | [127] |

Atria | $52.5\pm 3.1$ | $177.5\pm 9.0$ | - | $109.6\pm 3.6$ | $110.2\pm 84.0$ | - | [128] |

Atria | - | - | - | $85.0\pm 5.5$ | $66.1\pm 5.9$ | - | [129] |

Atria | - | - | - | $88.3\pm 2.5$ | $73.3\pm 1.7$ | - | [129] |

Atria | Ventricle | |||
---|---|---|---|---|

Parameter | Original Value | Optimized | Original Value | Optimized |

${k}_{\mathrm{u}}$ | 1/ms | 1/ms | 1/ms | 0.04/ms |

${n}_{\mathrm{Tm}}$ | 5 | 5 | 5 | 2.4 |

${\left[{\mathrm{Ca}}^{2+}\right]}_{\mathrm{T}50}^{\mathrm{ref}}$ | 0.86 µM | 1.05 µM | 0.805 µM | 1.05 µM |

${\beta}_{1}$ | −2.4 | −0.5 | −2.4 | −2.4 |

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Gerach, T.; Schuler, S.; Fröhlich, J.; Lindner, L.; Kovacheva, E.; Moss, R.; Wülfers, E.M.; Seemann, G.; Wieners, C.; Loewe, A.
Electro-Mechanical Whole-Heart Digital Twins: A Fully Coupled Multi-Physics Approach. *Mathematics* **2021**, *9*, 1247.
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**AMA Style**

Gerach T, Schuler S, Fröhlich J, Lindner L, Kovacheva E, Moss R, Wülfers EM, Seemann G, Wieners C, Loewe A.
Electro-Mechanical Whole-Heart Digital Twins: A Fully Coupled Multi-Physics Approach. *Mathematics*. 2021; 9(11):1247.
https://doi.org/10.3390/math9111247

**Chicago/Turabian Style**

Gerach, Tobias, Steffen Schuler, Jonathan Fröhlich, Laura Lindner, Ekaterina Kovacheva, Robin Moss, Eike Moritz Wülfers, Gunnar Seemann, Christian Wieners, and Axel Loewe.
2021. "Electro-Mechanical Whole-Heart Digital Twins: A Fully Coupled Multi-Physics Approach" *Mathematics* 9, no. 11: 1247.
https://doi.org/10.3390/math9111247