A Distributed Lumped Parameter Model of Blood Flow

Abstract

We propose a distributed lumped parameter (DLP) modeling framework to efficiently compute blood flow and pressure in vascular domains. This is achieved by developing analytical expressions describing expected energy losses along vascular segments, including from viscous dissipation, unsteadiness, flow separation, vessel curvature and vessel bifurcations. We apply this methodology to solve for unsteady blood flow and pressure in a variety of complex 3D image-based vascular geometries, which are typically approached using computational fluid dynamics (CFD) simulations. The proposed DLP framework demonstrated consistent agreement with CFD simulations in terms of flow rate and pressure distribution, with mean errors less than 7% over a broad range of hemodynamic conditions and vascular geometries. The computational cost of the DLP framework is orders of magnitude lower than the computational cost of CFD, which opens new possibilities for hemodynamics modeling in timely decision support scenarios, and a multitude of applications of imaged-based modeling that require ensembles of numerical simulations.

Appendices

Appendix A: Model Boundary Conditions

This section describes the specification of boundary conditions for all models. Further description on the boundary conditions, as well as imaging and available model data, are posted at www.vascularmodel.com. All of the CAD models are available at this site, except the four coronary models, which are uploaded as part of the electronic supplementary material.

Aortic Models

Phase-contrast (PC) MRI was used to measure volumetric flow in the ascending aorta and the respective waveforms for each patient (see inset panels in Fig. 2I) was mapped to the inlet of the CFD models using a time varying parabolic flow profile as the inflow boundary condition. Three-element “RCR” Windkessel models were coupled at all outlets. Information regarding how to choose the RCR parameters for each outlet are detailed in LaDisa et al.28 Regional mesh refinement was in the aortic coarctation models to resolve the complex flow features in the stenotic region.

Aorto-femoral Models

For Model A, an aortic flow waveform was adapted from Olufsen et al. 35 to have a mean cardiac output of 4.6 L/min (female). For Model B the supraceliac aorta blood flow waveform was derived from PC-MRI. For Models C and D individualized inflow boundary conditions were determined based on the Baker equation,5 relating body surface area to cardiac output, and assuming that $\sim \,70\%$ of the cardiac output is distributed to the supraceliac aorta.36 The resulting mean flows were used to generate inflow waveforms by scaling a gender-matched representative supraceliac aortic flow waveform. RCR models were applied at each outlet. The RCR parameters for each outlet were determined based on flow distributions to the outlets obtained from clinical PC-MRI measurements for Model B, or from literature data33 for Models A, C and D.

Coronary Models

In all cases, aortic flow was prescribed at the inlet, an RCR of the systemic circulation was coupled at the aortic outlet, and coronary-specific LPNs (see Fig. 6I) that consider the time-dependent intramyocardial pressure were coupled at each coronary outlet (separate LPN for each outlet). The effect of intramyocardial pressure is modeled by scaling the typical left and right ventricular pressures to recover realistic coronary flow waveforms. The LPN parameters of the systemic and coronary outlets were tuned to match target pressure and flow splits to the aorta and systemic and coronary outlets. A detailed description of the tuning procedure is given in Sankaran et al.38 Mesh refinement was used in all cases with stenotic lesions.

Cerebrovascular Models

For all models, a characteristic vertebral blood flow waveform from the literature18 was scaled to match time-averaged PC-MRI measurements in the vertebral arteries and mapped to a time-varying parabolic profile at the model inlet. Resistance boundary conditions were used at the outlets. A total resistance was calculated and distributed amongst the outlets by assuming all outlets act in parallel with resistance values inversely proportional to the outlet area. More details on boundary conditions are given in Bockman et al.11

Pulmonary Models

Pulmonary blood flow waveforms from PC-MRI were applied to the inlet of each model. The inflow waveforms were manipulated to have zero back flow to avoid numerical instability in the CFD simulations. Resistance values were assigned at the outlets based on the estimated mean pressure values for each patient, cross sectional area of the outlets, and left to right pulmonary flow split ratio obtained from PC-MRI data. Detailed description of resistance tuning for the pulmonary modeling is given in Tang et al.44

Congenital Heart Disease Models

PC-MRI data was used to prescribe an inflow waveform to the inlets of computational domains. Inflow waveforms prescribed at the inferior and superior vena cava (IVC, SVC), internal jugular vein (IJV) and broncheocephilic vein (BrS) are shown in Fig. 12I for all models. RCR models were coupled to each outlet, which parameters tuned to match target pressure values and assuming the LPA/RPA flow split ratio to be 45/55 for all patients. All numerical values for boundary condition parameters are available at www.vascularmodel.com.

Appendix B: Representative Example of DLP Modeling Results

A representative example of the computed resistance values from DLP modeling are presented for better understanding the relative contribution of various sources of energy dissipation. Table 1 shows resistance values at systole determined from the patient-specific coronary simulation shown in Fig. 7.

Table 1 Resistance values in cgs units ($g/s\cdot cm^4$) at systole from DLP modeling for patient-specific coronary simulation shown in Fig. 7.