Large vessels as a tree of transmission lines incorporated in the CircAdapt whole-heart model: A computational tool to examine heart-vessel interaction

Abstract

We developed a whole-circulation computational model by integrating a transmission line (TL) model describing vascular wave transmission into the established CircAdapt platform of whole-heart mechanics. In the present paper, we verify the numerical framework of our TL model by benchmark comparison to a previously validated pulse wave propagation (PWP) model. Additionally, we showcase the integrated CircAdapt-TL model, which now includes the heart as well as extensive arterial and venous trees with terminal impedances. We present CircAdapt-TL haemodynamics simulations of: 1) a systemic normotensive situation and 2) a systemic hypertensive situation. In the TL-PWP benchmark comparison we found good agreement regarding pressure and flow waveforms (relative errors ≤ 2.9% for pressure, and ≤ 5.6% for flow). CircAdapt-TL simulations reproduced the typically observed haemodynamic changes with hypertension, expressed by increases in mean and pulsatile blood pressures, and increased arterial pulse wave velocity. We observed a change in the timing of pressure augmentation (defined as a late-systolic boost in aortic pressure) from occurring after time of peak systolic pressure in the normotensive situation, to occurring prior to time of peak pressure in the hypertensive situation. The pressure augmentation could not be observed when the systemic circulation was lumped into a (non-linear) three-element windkessel model, instead of using our TL model. Wave intensity analysis at the carotid artery indicated earlier arrival of reflected waves with hypertension as compared to normotension, in good qualitative agreement with findings in patients. In conclusion, we successfully embedded a TL model as a vascular module into the CircAdapt platform. The integrated CircAdapt-TL model allows detailed studies on mechanistic studies on heart-vessel interaction.

Fig 1. The new vascular module interfaces the existing whole heart-heart mechanics and arterio-venous impedance modules.

The CircAdapt–TL model contains the cardiac modules describing whole-heart mechanics, including interventricular interactions, and the systemic and pulmonary circulations [9]. Cardiac valves are modelled as described in Palau-Caballero et al. [1]. The vascular module is described in section ‘Vascular module’. The arterio-venous impedance module, modelling the peripheral circulation using a non-linear three-element windkessel (3WK) was previously developed by Arts et al. [10]. Abbreviations: RA: right atrium, LA: left atrium, TV: tricuspid valve, PV: pulmonary valve, AV: aortic valve, RV: right ventricle, LV: left ventricle, MV: mitral valve, art: arterial, ven: venous, 3WK: non-linear three-element windkessel model. Symbols: R_p: peripheral resistance, Z_wave: wave impedance, C: compliance.

Fig 2. Overview of the complete arterial model domain.

Geometrical and mechanical properties of modelled arteries are given in Table B in S1 Text. Panels display pressure and flow waveforms for three regions (i.e. central, arm and leg, indicated using symbol and colour coding). Grey curves display left ventricular (LV) pressures and volumes, left atrial (LA) pressures and volumes as well as aortic valve flow. Line type indicates the distance from the heart within a region; proximal: solid line, intermediate: dash-dotted line, distal: dotted line, respectively. Arrows indicate the direction of mean blood flow. Aortic valve closure is indicated by the vertical blue dashed lines.

Fig 3. Overview of the complete venous model domain.

Geometrical and mechanical properties of modelled veins are given in Table C in S1 Text. Panels display pressure and flow waveforms for three regions; central, arm and leg, indicated using symbol and colour coding. Line type indicates the distance from the heart within a region; proximal: solid line, intermediate: dash-dotted line, distal: dotted line, respectively. Grey curves display right ventricular (RV) pressures and volumes, right atrial (RA) pressures and volumes. Arrows indicate the direction of mean blood flow. Aortic valve closure is indicated by the vertical blue dashed lines.

Fig 4. Results of the benchmark comparison.

Pressure (p) and flow (q) waveforms generated by the transmission line (TL) model and 1D pulse wave propagation model (PWP) model are shown for various locations along the arterial domain. Agreement between p and q waveforms is expressed by root mean square errors ϵ_p and ϵ_q, as well as relative errors δ_p and δ_q.

Fig 5. CircAdapt–TL model simulated time courses of pressure and flow for the reference (REF–TL) simulation (left), and the hypertension (HYP–TL) simulation (right).

Curves display time courses of pressure (solid lines) and flow (dotted lines) of the left ventricle and large arteries; the right ventricle and pulmonary artery; the left atrium and pulmonary vein; and the right atrium and vena cava, respectively. Valve states (i.e. open or closed) are indicated. These have been determined exactly from the valve model orifice area data, not from the pressure or flow signals. AVC: aortic valve closed, AVO: aortic valve open, MVC: mitral valve closed, MVO: mitral valve open, PVC: pulmonary valve closed, PVO: pulmonary valve open, TVC: tricuspid valve closed, TVO: tricuspid valve open.

Fig 6. Left ventricular- and aortic blood pressure tracings simulated using, on the one hand, the CircAdapt model with the systemic circulation lumped into a non-linear three-element windkessel model (3WK) and, on the other hand, CircAdapt with the systemic circulation represented by transmission lines (TL).

Pressure tracings are given for the normotensive (REF–3WK and REF–TL, respectively) and hypertensive situation (HYP–3WK and HYP–TL, respectively).

Fig 7. Wave intensity analysis.

A: Carotid arterial pressure- and flow velocity waveforms of the reference (REF–TL) and hypertensive (HYP–TL) simulations, and corresponding wave intensity traces. B: Normalised carotid arterial wave intensity traces of two patients (adapted from [31], with reference to type C- and A-waveforms). Abbreviations: dI⁺, dI⁻ and dI denote forward-, backward-, and net wave intensity, respectively. FCW, BCW, FEW: forward compression wave, backward compression wave, and forward expansion wave, respectively.