Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and kernel methods

Abstract

In this work, we consider 2 kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery, i.e., in an artery that is located far away from the heart. For our simulations, we place the stenosis in one of the tibial arteries belonging to the right lower leg (right posterior tibial artery). The model reduction techniques that are used are on the one hand dimensionally reduced models (1-D and 0-D models, the so-called mixed-dimension model) and on the other hand surrogate models produced by kernel methods. Both methods are combined in such a way that the mixed-dimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a well-trained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation or state estimation problems can be solved in a very efficient way. Furthermore, it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow, obtaining a speedup of several orders over the full model.

Deep learning for prediction of cardiac indices from photoplethysmographic waveform: A virtual database approach

“…Overfitting could perhaps be avoided by generating (significantly) more data, but this was not possible due to relatively high computational cost to run the cardiovascular simulator. Developing more computationally efficient cardiovascular models (eg, using approach presented in Koepll et al) could perhaps be helpful in this issue.…”

Mathematical modelling of atherosclerosis

“…u are constants, ρ is blood density, R is diameter of the vessel. Formula (4.4) is considered in [52], (4.5) in [116], (4.6) in [94,109,156,182], (4.7) in [137]. The case of several identical stenoses in one vessel is considered in [151], and two subsequent stenoses with different lesions is considered in [14].…”

A 3D‐1D coupled blood flow and oxygen transport model to generate microvascular networks

“…Numerical simulation techniques have become an important part in testing hypotheses when experimental investigations are not possible or are limited by accessibility, size and resolution. [3][4][5][6][7][8][9] To be able to perform conclusive blood flow simulations, it is crucial to have a precise description of the network. This means that accurate data on the radii, lengths and connectivity of the vessels as well as the location of the vessels are required.…”

“…Moreover, a reliable and robust blood flow model allows one to investigate the impact of diseases and therapeutic procedures in a non‐invasive way. Numerical simulation techniques have become an important part in testing hypotheses when experimental investigations are not possible or are limited by accessibility, size and resolution 3‐9 . To be able to perform conclusive blood flow simulations, it is crucial to have a precise description of the network.…”