Transient integral boundary layer method to calculate the translesional pressure drop and the fractional flow reserve in myocardial bridges

BACKGROUND: The pressure drop – flow relations in myocardial bridges and the assessment of vascular heart disease via fractional flow reserve (FFR) have motivated many researchers the last decades. The aim of this study is to simulate several clinical conditions present in myocardial bridges to determine the flow reserve and consequently the clinical relevance of the disease. From a fluid mechanical point of view the pathophysiological situation in myocardial bridges involves fluid flow in a time dependent flow geometry, caused by contracting cardiac muscles overlying an intramural segment of the coronary artery. These flows mostly involve flow separation and secondary motions, which are difficult to calculate and analyse. METHODS: Because a three dimensional simulation of the haemodynamic conditions in myocardial bridges in a network of coronary arteries is time-consuming, we present a boundary layer model for the calculation of the pressure drop and flow separation. The approach is based on the assumption that the flow can be sufficiently well described by the interaction of an inviscid core and a viscous boundary layer. Under the assumption that the idealised flow through a constriction is given by near-equilibrium velocity profiles of the Falkner-Skan-Cooke (FSC) family, the evolution of the boundary layer is obtained by the simultaneous solution of the Falkner-Skan equation and the transient von-Kármán integral momentum equation. RESULTS: The model was used to investigate the relative importance of several physical parameters present in myocardial bridges. Results have been obtained for steady and unsteady flow through vessels with 0 – 85% diameter stenosis. We compare two clinical relevant cases of a myocardial bridge in the middle segment of the left anterior descending coronary artery (LAD). The pressure derived FFR of fixed and dynamic lesions has shown that the flow is less affected in the dynamic case, because the distal pressure partially recovers during re-opening of the vessel in diastole. We have further calculated the wall shear stress (WSS) distributions in addition to the location and length of the flow reversal zones in dependence on the severity of the disease. CONCLUSION: The described boundary layer method can be used to simulate frictional forces and wall shear stresses in the entrance region of vessels. Earlier models are supplemented by the viscous effects in a quasi three-dimensional vessel geometry with a prescribed wall motion. The results indicate that the translesional pressure drop and the mean FFR compares favourably to clinical findings in the literature. We have further shown that the mean FFR under the assumption of Hagen-Poiseuille flow is overestimated in developing flow conditions

Forward modeling and tissue conductivities

The neuroelectromagnetic forward model describes the prediction of measurements from known sources. It includes models for the sources and the sensors as well as an electromagnetic description of the head as a volume conductor, which are discussed in this chapter. First we give a general overview on the forward problem and discuss various simplifications and assumptions that lead to different analytical and numerical methods. Next, we introduce important analytical models which assume simple geometries of the head. Then we describe numerical models accounting for realistic geometries. The most important numerical methods for head modeling are the boundary element method (BEM) and the finite element method (FEM). The boundary element method describes the head by a small number of compartments, each with a homogeneous isotropic conductivity. In contrast, the finite element method discretizes the 3D distribution of the anisotropic conductivity tensor with the help of small volume elements. Subsequently, we discuss in some detail how electrical conductivity information is measured and how it is used in forward modeling. Finally, we briefly introduce the lead field concept

Forward modeling and tissue conductivities

The neuroelectromagnetic forward model describes the prediction of measurements from known sources. It includes models for the sources and the sensors as well as an electromagnetic description of the head as a volume conductor, which are discussed in this chapter. First we give a general overview on the forward problem and discuss various simplifications and assumptions that lead to different analytical and numerical methods. Next, we introduce important analytical models which assume simple geometries of the head. Then we describe numerical models accounting for realistic geometries. The most important numerical methods for head modeling are the boundary element method (BEM) and the finite element method (FEM). The boundary element method describes the head by a small number of compartments, each with a homogeneous isotropic conductivity. In contrast, the finite element method discretizes the 3D distribution of the anisotropic conductivity tensor with the help of small-volume elements. Subsequently, we discuss in some detail how electrical conductivity information is measured and how it is used in forward modeling. Finally, we briefly introduce the lead field concept