Optimization of topological complexity for one-dimensional arterial blood flow models

Figure 1.. Two baseline models were used in this work: a model containing 96 arterial segments in which parameters and topology were adapted from [9] (a), and a patient-specific coronary network (b). The arrow indicates the location of invasive pressure measurements, and the section coloured in red is a significant stenosis.

Figure 2.. An example of a reduced network that has been obtained from the baseline model in figure 1 by lumping 1D model segments into WK3 models. Lumped 1D model segments are shown in colour.

Figure 3.. Illustration of an arterial 1D model segment coupled to a WK3 model that may be lumped into an equivalent compliance Ct according to equation (2.10). (Online version in colour.)

Figure 4.. Comparison of Method 1 and Method 2 for network reduction. Baseline model in black and reduced models in red together with pressure and flow waveforms at the inlet of the right internal carotid artery. Impedance modulus and phase angle are also shown.

Figure 5.. Optimal networks and corresponding waveforms (solid lines) obtained from the 96-artery baseline model with pressure at midpoint of ascending aorta (left) and right brachial artery (middle) set as quantities of interest. In the top and middle rows, an averaged (of the two quantities of interest) error threshold of εP,sys + εP,dia less than 0.3% (top row) and 1.0% (middle row) was used. In the last case, an error threshold of εP,aug + εPP less than 0.7% was used for the aorta. Dashed waveforms correspond to simulations of normal ageing as described in §2.6.

Figure 6.. Optimal networks obtained from the 96-artery baseline model with the flow at the distal end of the right carotid artery set as quantity of interest (left panel) and with 0.9 and 3.4% average error thresholds, respectively. The right panel show results when the flow at the proximal end of the right middle cerebral artery was set as quantity of interest and with 0.6 and 1.6% average error thresholds, respectively.

Figure 7.. Optimal networks obtained from the coronary network with difference in iFR set as quantity of interest, with εiFR < 0.00012 (top) and εiFR < 0.04 (bottom). Results are visualized through the distal pressure waveform. In vivo pressure waveform and predicted velocity waveforms are also shown. The arrow indicates the location of the waveforms, and the red section a significant stenosis.

Figure 8.. Proximal Pp and distal Pd pressure waveforms obtained during invasive angiography, together with a computed left ventricle pressure waveform in grey. The last 30% of the cardiac cycle is highlighted in red and was used to compute the instantaneous wave-free ratio, iFR, a drug-free index of the significance of the stenosis.

Figure 9.. Schematic of the two lumped parameter models used in this work, WK3 model (a) and WKcor (b). R1, R2, Rp, Rm and Rd are resistance parameters, C, Ca and Cm are compliance parameters and PLV and Pout,WK are left ventricle and outflow windkessel pressures, respectively.

Figure 10.. Arrows indicate the direction (not magnitude) of Q¯, and also which arteries are lumped into WK3 models for two selected sites of truncation: the left and right internal carotid arteries.

Figure 11.. Compliance contribution from vessels in bifurcations and anastomosis used in equation (A6a) and equation (A6b). Arrows indicate the direction of flow.

Figure 12.. Illustration of vessels distal of points of truncation lumped into WK3 models.