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**A 1D arterial blood flow model incorporating ventricular pressure, aortic valve and regional coronary flow using the locally conservative Galerkin (LCG) method.**
*(English)*
Zbl 1137.92009

Summary: There is an important interaction between the pumping performance of the ventricle, arterial haemodynamics and coronary blood flow. While previous nonlinear 1D models have focused only on one of these components, the model presented in this study includes coronary and systemic arterial circulations, as well as ventricular pressure and an aortic valve that opens and closes ‘independently’ and based on local haemodynamics. The systemic circulation is modelled as a branching network of elastic tapering vessels. The terminal element applied at the extremities of the network is a single tapering vessel which is shown to adequately represent the input characteristics of the downstream vasculature. The coronary model consists of left and right coronary arteries which both branch into two ‘equivalent’ vessels that account for the lumped characteristics of subendocardial and subepicardial flows. As the contracting heart muscle causes significant compression of the subendocardial vessels, a time-varying external pressure proportional to the ventricular pressure is applied to the distal part of the equivalent subendocardial vessel. The aortic valve is modelled using a variable reflection coefficient with respect to backward-running aortic waves, and a variable transmission coefficient with respect to forward-running ventricular waves.

A realistic ventricular pressure is the input to the system; however, an afterload-corrected ventricular pressure is calculated and results in pressure gradients between the ventricle and aorta that are similar to those observed in vivo. The 1D equations of fluid flow are solved using the locally conservative Galerkin method, which provides explicit element-wise conservation, and can naturally incorporate vessel branching. Each component of the model is verified using a number of tests to ensure accuracy and reveal the underlying processes that give rise to complex pressure and flow waveforms. The complete model is then implemented, and simulations are performed with input parameters representing ‘at rest’ and exercise states for a normal adult. The resulting waveforms contain all of the important features seen in vivo, and standard measures of haemodynamic state are found to be normal. In addition, one or several characteristics of some common diseases are imposed on the model and are found to produce haemodynamic changes that agree with experimental observations in the published literature. Using a patient-specific carotid bifurcation geometry, 1D velocity waveforms are also compared with waveforms obtained from a three-dimensional model. The 1D and 3D results show good agreement.

A realistic ventricular pressure is the input to the system; however, an afterload-corrected ventricular pressure is calculated and results in pressure gradients between the ventricle and aorta that are similar to those observed in vivo. The 1D equations of fluid flow are solved using the locally conservative Galerkin method, which provides explicit element-wise conservation, and can naturally incorporate vessel branching. Each component of the model is verified using a number of tests to ensure accuracy and reveal the underlying processes that give rise to complex pressure and flow waveforms. The complete model is then implemented, and simulations are performed with input parameters representing ‘at rest’ and exercise states for a normal adult. The resulting waveforms contain all of the important features seen in vivo, and standard measures of haemodynamic state are found to be normal. In addition, one or several characteristics of some common diseases are imposed on the model and are found to produce haemodynamic changes that agree with experimental observations in the published literature. Using a patient-specific carotid bifurcation geometry, 1D velocity waveforms are also compared with waveforms obtained from a three-dimensional model. The 1D and 3D results show good agreement.

### MSC:

92C35 | Physiological flow |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

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\textit{J. P. Mynard} and \textit{P. Nithiarasu}, Commun. Numer. Methods Eng. 24, No. 5, 367--417 (2008; Zbl 1137.92009)

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