A unified method for estimating pressure losses at vascular junctions

A one-dimensional computational model for blood flow in an elastic blood vessel with a rigid catheter

Strokes are one of the leading causes of death in the United States. Stroke treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. A computer simulation to systematically plan such patient-specific treatments needs a network of about 105 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time. This work presents a novel one-dimensional mathematical formulation for blood flow modeling in an elastic blood vessel with a centrally placed rigid catheter. The governing equations are first-order hyperbolic partial differential equations, and the hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We employed the Discontinuous Galerkin method to solve the hyperbolic system and validated the implementation by comparing it against a well-established 3D CFD solver using idealized vessels and a realistic truncated arterial network. The results showed clinically insignificant differences in steady flow cases, with overall variations between 1D and 3D models remaining below 10%. Additionally, the solver accurately captured wave reflection phenomena at domain discontinuities in unsteady cases. A primary advantage of this model over 3D solvers is its ease in obtaining a discretized geometry of complex vasculatures with multiple arterial branches. Thus, the 1D computational model offers good accuracy and applicability in simulating complex vasculatures, demonstrating promising potential for investigating patient-specific endovascular interventions in strokes.

A computational study of aortic reconstruction in single ventricle patients

Patients with hypoplastic left heart syndrome (HLHS) are born with an underdeveloped left heart. They typically receive a sequence of surgeries that result in a single ventricle physiology called the Fontan circulation. While these patients usually survive into early adulthood, they are at risk for medical complications, partially due to their lower than normal cardiac output, which leads to insufficient cerebral and gut perfusion. While clinical imaging data can provide detailed insight into cardiovascular function within the imaged region, it is difficult to use these data for assessing deficiencies in the rest of the body and for deriving blood pressure dynamics. Data from patients used in this paper include three-dimensional, magnetic resonance angiograms (MRA), time-resolved phase contrast cardiac magnetic resonance images (4D-MRI) and sphygmomanometer blood pressure measurements. The 4D-MRI images provide detailed insight into velocity and flow in vessels within the imaged region, but they cannot predict flow in the rest of the body, nor do they provide values of blood pressure. To remedy these limitations, this study combines the MRA, 4D-MRI, and pressure data with 1D fluid dynamics models to predict hemodynamics in the major systemic arteries, including the cerebral and gut vasculature. A specific focus is placed on studying the impact of aortic reconstruction occurring during the first surgery that results in abnormal vessel morphology. To study these effects, we compare simulations for an HLHS patient with simulations for a matched control patient that has double outlet right ventricle (DORV) physiology with a native aorta. Our results show that the HLHS patient has hypertensive pressures in the brain as well as reduced flow to the gut. Wave intensity analysis suggests that the HLHS patient has irregular circulatory function during light upright exercise conditions and that predicted wall shear stresses are lower than normal, suggesting the HLHS patient may have hypertension.

Automated generation of 0D and 1D reduced-order models of patient-specific blood flow

Three-dimensional (3D) cardiovascular fluid dynamics simulations typically require hours to days of computing time on a high-performance computing cluster. One-dimensional (1D) and lumped-parameter zero-dimensional (0D) models show great promise for accurately predicting blood bulk flow and pressure waveforms with only a fraction of the cost. They can also accelerate uncertainty quantification, optimization, and design parameterization studies. Despite several prior studies generating 1D and 0D models and comparing them to 3D solutions, these were typically limited to either 1D or 0D and a singular category of vascular anatomies. This work proposes a fully automated and openly available framework to generate and simulate 1D and 0D models from 3D patient-specific geometries, automatically detecting vessel junctions and stenosis segments. Our only input is the 3D geometry; we do not use any prior knowledge from 3D simulations. All computational tools presented in this work are implemented in the open-source software platform SimVascular. We demonstrate the reduced-order approximation quality against rigid-wall 3D solutions in a comprehensive comparison with N = 72 publicly available models from various anatomies, vessel types, and disease conditions. Relative average approximation errors of flows and pressures typically ranged from 1% to 10% for both 1D and 0D models, measured at the outlets of terminal vessel branches. In general, 0D model errors were only slightly higher than 1D model errors despite requiring only a third of the 1D runtime. Automatically generated ROMs can significantly speed up model development and shift the computational load from high-performance machines to personal computers.

A nonlinear multi-scale model for blood circulation in a realistic vascular system

In the last decade, numerical models have become an increasingly important tool in biological and medical science. Numerical simulations contribute to a deeper understanding of physiology and are a powerful tool for better diagnostics and treatment. In this paper, a nonlinear multi-scale model framework is developed for blood flow distribution in the full vascular system of an organ. We couple a quasi one-dimensional vascular graph model to represent blood flow in larger vessels and a porous media model to describe flow in smaller vessels and capillary bed. The vascular model is based on Poiseuille's Law, with pressure correction by elasticity and pressure drop estimation at vessels' junctions. The porous capillary bed is modelled as a two-compartment domain (artery and venous) using Darcy's Law. The fluid exchange between the artery and venous capillary bed compartments is defined as blood perfusion. The numerical experiments show that the proposed model for blood circulation: (i) is closely dependent on the structure and parameters of both the larger vessels and of the capillary bed, and (ii) provides a realistic blood circulation in the organ. The advantage of the proposed model is that it is complex enough to reliably capture the main underlying physiological function, yet highly flexible as it offers the possibility of incorporating various local effects. Furthermore, the numerical implementation of the model is straightforward and allows for simulations on a regular desktop computer.

High fidelity blood flow in a patient-specific arteriovenous fistula

An arteriovenous fistula, created by artificially connecting segments of a patient’s vasculature, is the preferred way to gain access to the bloodstream for kidney dialysis. The increasing power and availability of supercomputing infrastructure means that it is becoming more realistic to use simulations to help identify the best type and location of a fistula for a specific patient. We describe a 3D fistula model that uses the lattice Boltzmann method to simultaneously resolve blood flow in patient-specific arteries and veins. The simulations conducted here, comprising vasculatures of the whole forearm, demonstrate qualified validation against clinical data. Ongoing research to further encompass complex biophysics on realistic time scales will permit the use of human-scale physiological models for basic and clinical medicine.

Impact of tapering of arterial vessels on blood pressure, pulse wave velocity, and wave intensity analysis using one-dimensional computational model

The angle of arterial tapering increases with ageing, and the geometrical changes of the aorta may cause an increase in central arterial pressure and stiffness. The impact of tapering has been primarily studied using frequency-domain transmission line theories. In this work, we revisit the problem of tapering and investigate its effect on blood pressure and pulse wave velocity (PWV) using a time-domain analysis with a 1D computational model. First, tapering is modelled as a stepwise reduction in diameter and compared with results from a continuously tapered segment. Next, we studied wave reflections in a combination of stepwise diameter reduction of straight vessels and bifurcations, then repeated the experiments with decreasing the length to physiological values. As the model's segments became shorter in length, wave reflections and re-reflections resulted in waves overlapping in time. We extended our work by examining the effect of increasing the tapering angle on blood pressure and wave intensity in physiological models: a model of the thoracic aorta and a model of upper thoracic and descending aorta connected to the iliac bifurcation. Vessels tapering inherently changed the ratio between the inlet and outlet cross-sectional areas, increasing the vessel resistance and reducing the compliance compared with non-tapered vessels. These variables influence peak and pulse pressure. In addition, it is well established that pulse wave velocity increases in an ageing arterial tree. This work provides confirmation that tapering induces reflections and offers an additional explanation to the observation of increased peak pressure and decreased diastolic pressure distally in the arterial tree.

Accelerated Estimation of Pulmonary Artery Stenosis Pressure Gradients with Distributed Lumped Parameter Modeling vs. 3D CFD with Instantaneous Adaptive Mesh Refinement: Experimental Validation in Swine

Branch pulmonary artery stenosis (PAS) commonly occurs in congenital heart disease and the pressure gradient over a stenotic PA lesion is an important marker for re-intervention. Image based computational fluid dynamics (CFD) has shown promise for non-invasively estimating pressure gradients but one limitation of CFD is long simulation times. The goal of this study was to compare accelerated predictions of PAS pressure gradients from 3D CFD with instantaneous adaptive mesh refinement (AMR) versus a recently developed 0D distributed lumped parameter CFD model. Predictions were then experimentally validated using a swine PAS model (n = 13). 3D CFD simulations with AMR improved efficiency by 5 times compared to fixed grid CFD simulations. 0D simulations further improved efficiency by 6 times compared to the 3D simulations with AMR. Both 0D and 3D simulations underestimated the pressure gradients measured by catheterization (- 1.87 ± 4.20 and - 1.78 ± 3.70 mmHg respectively). This was partially due to simulations neglecting the effects of a catheter in the stenosis. There was good agreement between 0D and 3D simulations (ICC 0.88 [0.66-0.96]) but only moderate agreement between simulations and experimental measurements (0D ICC 0.60 [0.11-0.86] and 3D ICC 0.66 [0.21-0.88]). Uncertainty assessment indicates that this was likely due to limited medical imaging resolution causing uncertainty in the segmented stenosis diameter in addition to uncertainty in the outlet resistances. This study showed that 0D lumped parameter models and 3D CFD with instantaneous AMR both improve the efficiency of hemodynamic modeling, but uncertainty from medical imaging resolution will limit the accuracy of pressure gradient estimations.

A Distributed Lumped Parameter Model of Blood Flow

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.

Non-invasive coronary CT angiography-derived fractional flow reserve: A benchmark study comparing the diagnostic performance of four different computational methodologies

Non-invasive coronary computed tomography (CT) angiography-derived fractional flow reserve (cFFR) is an emergent approach to determine the functional relevance of obstructive coronary lesions. Its feasibility and diagnostic performance has been reported in several studies. It is unclear if differences in sensitivity and specificity between these studies are due to study design, population, or "computational methodology." We evaluate the diagnostic performance of four different computational workflows for the prediction of cFFR using a limited data set of 10 patients, three based on reduced-order modelling and one based on a 3D rigid-wall model. The results for three of these methodologies yield similar accuracy of 6.5% to 10.5% mean absolute difference between computed and measured FFR. The main aspects of modelling which affected cFFR estimation were choice of inlet and outlet boundary conditions and estimation of flow distribution in the coronary network. One of the reduced-order models showed the lowest overall deviation from the clinical FFR measurements, indicating that reduced-order models are capable of a similar level of accuracy to a 3D model. In addition, this reduced-order model did not include a lumped pressure-drop model for a stenosis, which implies that the additional effort of isolating a stenosis and inserting a pressure-drop element in the spatial mesh may not be required for FFR estimation. The present benchmark study is the first of this kind, in which we attempt to homogenize the data required to compute FFR using mathematical models. The clinical data utilised in the cFFR workflows are made publicly available online.

A novel, FFT-based one-dimensional blood flow solution method for arterial network

In the present work, we propose an FFT-based method for solving blood flow equations in an arterial network with variable properties and geometrical changes. An essential advantage of this approach is in correctly accounting for the vessel skin friction through the use of Womersley solution. To incorporate nonlinear effects, a novel approximation method is proposed to enable calculation of nonlinear corrections. Unlike similar methods available in the literature, the set of algebraic equations required for every harmonic is constructed automatically. The result is a generalized, robust and fast method to accurately capture the increasing pulse wave velocity downstream as well as steepening of the pulse front. The proposed method is shown to be appropriate for incorporating correct convection and diffusion coefficients. We show that the proposed method is fast and accurate and it can be an effective tool for 1D modelling of blood flow in human arterial networks.

High-Frequency Fluctuations in Post-stenotic Patient Specific Carotid Stenosis Fluid Dynamics: A Computational Fluid Dynamics Strategy Study

Purpose

Screening of asymptomatic carotid stenoses is performed by auscultation of the carotid bruit, but the sensitivity is poor. Instead, it has been suggested to detect carotid bruit as neck’s skin vibrations. We here take a first step towards a computational fluid dynamics proof-of-concept study, and investigate the robustness of our numerical approach for capturing high-frequent fluctuations in the post-stenotic flow. The aim was to find an ideal solution strategy from a pragmatic point of view, balancing accuracy with computational cost comparing an under-resolved direct numerical simulation (DNS) approach vs. three common large eddy simulation (LES) models (static/dynamic Smagorinsky and Sigma).

Method

We found a reference solution by performing a spatial and temporal refinement study of a stenosed carotid bifurcation with constant flow rate. The reference solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\Delta x = 1.92 \times 10^{ - 4} \;{\text{m}},\; \Delta t = 5 \times 10^{ - 5} \;{\text{s}}} \right)$$\end{document}Δx=1.92×10-4m,Δt=5×10-5s was compared against LES for both a constant and pulsatile flow.

Results

Only the Sigma and Dynamic Smagorinsky models were able to replicate the flow field of the reference solution for a pulsatile simulation, however the computational cost of the Sigma model was lower. However, none of the sub-grid scale models were able to replicate the high-frequent flow in the peak-systolic constant flow rate simulations, which had a higher mean Reynolds number.

Conclusions

The Sigma model was the best combination between accuracy and cost for simulating the pulsatile post-stenotic flow field, whereas for the constant flow rate, the under-resolved DNS approach was better. These results can be used as a reference for future studies investigating high-frequent flow features.

A novel porous media-based approach to outflow boundary resistances of 1D arterial blood flow models

In this paper we introduce a novel method for prescribing terminal boundary conditions in one-dimensional arterial flow networks. This is carried out by coupling the terminal arterial vessel with a poro-elastic tube, representing the flow resistance offered by microcirculation. The performance of the proposed porous media-based model has been investigated through several different numerical examples. First, we investigate model parameters that have a profound influence on the flow and pressure distributions of the system. The simulation results have been compared against the waveforms generated by three elements (RCR) Windkessel model. The proposed model is also integrated into a realistic arterial tree, and the results obtained have been compared against experimental data at different locations of the network. The accuracy and simplicity of the proposed model demonstrates that it can be an excellent alternative for the existing models.

A methodology for numerically analysing the hepatic artery haemodynamics during B-TACE: a proof of concept

Balloon-occluded transarterial chemoembolisation (B-TACE) is an intraarterial transcatheter treatment for liver cancer. In B-TACE, an artery-occluding microballoon catheter occludes an artery and promotes collateral circulation for drug delivery to tumours. This paper presents a methodology for analysing the haemodynamics during B-TACE, by combining zero-dimensional and three-dimensional modelling tools. As a proof of concept, we apply the methodology to a patient-specific hepatic artery geometry and analyse two catheter locations. Results show that the blood flow redistribution can be predicted in this proof-of-concept study, suggesting that this approach could potentially be used to optimise catheter location.

The FDA nozzle benchmark: "In theory there is no difference between theory and practice, but in practice there is"

The utility of flow simulations relies on the robustness of computational fluid dynamics (CFD) solvers and reproducibility of results. The aim of this study was to validate the Oasis CFD solver against in vitro experimental measurements of jet breakdown location from the FDA nozzle benchmark at Reynolds number 3500, which is in the particularly challenging transitional regime. Simulations were performed on meshes consisting of 5, 10, 17, and 28 million (M) tetrahedra, with Δt  =  10^-5 seconds. The 5M and 10M simulation jets broke down in reasonable agreement with the experiments. However, the 17M and 28M simulation jets broke down further downstream. But which of our simulations are "correct"? From a theoretical point of view, they are all wrong because the jet should not break down in the absence of disturbances. The geometry is axisymmetric with no geometrical features that can generate angular velocities. A stable flow was supported by linear stability analysis. From a physical point of view, a finite amount of "noise" will always be present in experiments, which lowers transition point. To replicate noise numerically, we prescribed minor random angular velocities (approximately 0.31%), much smaller than the reported flow asymmetry (approximately 3%) and model accuracy (approximately 1%), at the inlet of the 17M simulation, which shifted the jet breakdown location closer to the measurements. Hence, the high-resolution simulations and "noise" experiment can potentially explain discrepancies in transition between sometimes "sterile" CFD and inherently noisy "ground truth" experiments. Thus, we have shown that numerical simulations can agree with experiments, but for the wrong reasons.

Comparison of 1D and 3D Models for the Estimation of Fractional Flow Reserve

In this work we propose to validate the predictive capabilities of one-dimensional (1D) blood flow models with full three-dimensional (3D) models in the context of patient-specific coronary hemodynamics in hyperemic conditions. Such conditions mimic the state of coronary circulation during the acquisition of the Fractional Flow Reserve (FFR) index. Demonstrating that 1D models accurately reproduce FFR estimates obtained with 3D models has implications in the approach to computationally estimate FFR. To this end, a sample of 20 patients was employed from which 29 3D geometries of arterial trees were constructed, 9 obtained from coronary computed tomography angiography (CCTA) and 20 from intra-vascular ultrasound (IVUS). For each 3D arterial model, a 1D counterpart was generated. The same outflow and inlet pressure boundary conditions were applied to both (3D and 1D) models. In the 1D setting, pressure losses at stenoses and bifurcations were accounted for through specific lumped models. Comparisons between 1D models (FFR1D) and 3D models (FFR3D) were performed in terms of predicted FFR value. Compared to FFR3D, FFR1D resulted with a difference of 0.00 ± 0.03 and overall predictive capability AUC, Acc, Spe, Sen, PPV and NPV of 0.97, 0.98, 0.90, 0.99, 0.82, and 0.99, with an FFR threshold of 0.8. We conclude that inexpensive FFR1D simulations can be reliably used as a surrogate of demanding FFR3D computations.

Zero-dimensional lumped approach to incorporate the dynamic part of the pressure at vessel junctions in a 1D wave propagation model

A benchmark study by Boileau et al tested 6 commonly used numerical schemes for 1D wave propagation, for their ability to capture the main features of pressure, flow, and area waveforms in large arteries. While all numerical schemes showed good agreement in pressure and flow waveforms for smaller arterial networks, the simplified trapezium rule method proposed by Kroon et al showed an overestimation for the systolic pressure of 1% in proximal regions and an underestimation of 3% in distal regions in comparison with the 5 other schemes when using a larger arterial network, published as the ADAN56 model. The authors attributed this difference to the neglection of the dynamic part of the pressure at vessel junctions. Carson et al resolved these differences by proposing 2 methods to implement the dynamic part of the pressure in the simplified trapezium rule method scheme. In the present study, an alternative method is introduced extending the work by Kroon et al. This alternative method consists of a new 0D element, which is placed at vessel junctions. The strength of this new element is the ease of implementation and its flexible coupling with other elements, without introducing additional degrees of freedom or the need of a penalty function. This new approach is compared with 5 other numerical schemes, which already have the dynamic part of the pressure incorporated. The new method shows excellent agreement with these schemes for the ADAN56 model.

Roadmap for cardiovascular circulation model

Computational models of many aspects of the mammalian cardiovascular circulation have been developed. Indeed, along with orthopaedics, this area of physiology is one that has attracted much interest from engineers, presumably because the equations governing blood flow in the vascular system are well understood and can be solved with well-established numerical techniques. Unfortunately, there have been only a few attempts to create a comprehensive public domain resource for cardiovascular researchers. In this paper we propose a roadmap for developing an open source cardiovascular circulation model. The model should be registered to the musculo-skeletal system. The computational infrastructure for the cardiovascular model should provide for near real-time computation of blood flow and pressure in all parts of the body. The model should deal with vascular beds in all tissues, and the computational infrastructure for the model should provide links into CellML models of cell function and tissue function. In this work we review the literature associated with 1D blood flow modelling in the cardiovascular system, discuss model encoding standards, software and a model repository. We then describe the coordinate systems used to define the vascular geometry, derive the equations and discuss the implementation of these coupled equations in the open source computational software OpenCMISS. Finally, some preliminary results are presented and plans outlined for the next steps in the development of the model, the computational software and the graphical user interface for accessing the model.

Novel wave power analysis linking pressure-flow waves, wave potential, and the forward and backward components of hydraulic power

Wave intensity analysis provides detailed insights into factors influencing hemodynamics. However, wave intensity is not a conserved quantity, so it is sensitive to diameter variations and is not distributed among branches of a junction. Moreover, the fundamental relation between waves and hydraulic power is unclear. We, therefore, propose an alternative to wave intensity called “wave power,” calculated via incremental changes in pressure and flow (dPdQ) and a novel time-domain separation of hydraulic pressure power and kinetic power into forward and backward wave-related components (ΠP± and ΠQ±). Wave power has several useful properties: 1) it is obtained directly from flow measurements, without requiring further calculation of velocity; 2) it is a quasi-conserved quantity that may be used to study the relative distribution of waves at junctions; and 3) it has the units of power (Watts). We also uncover a simple relationship between wave power and changes in ΠP± and show that wave reflection reduces transmitted power. Absolute values of ΠP± represent wave potential, a recently introduced concept that unifies steady and pulsatile aspects of hemodynamics. We show that wave potential represents the hydraulic energy potential stored in a compliant pressurized vessel, with spatial gradients producing waves that transfer this energy. These techniques and principles are verified numerically and also experimentally with pressure/flow measurements in all branches of a central bifurcation in sheep, under a wide range of hemodynamic conditions. The proposed “wave power analysis,” encompassing wave power, wave potential, and wave separation of hydraulic power provides a potent time-domain approach for analyzing hemodynamics.

Non-dimensional physics of pulsatile cardiovascular networks and energy efficiency

In Nature, there exist a variety of cardiovascular circulation networks in which the energetic ventricular load has both steady and pulsatile components. Steady load is related to the mean cardiac output (CO) and the haemodynamic resistance of the peripheral vascular system. On the other hand, the pulsatile load is determined by the simultaneous pressure and flow waveforms at the ventricular outlet, which in turn are governed through arterial wave dynamics (transmission) and pulse decay characteristics (windkessel effect). Both the steady and pulsatile contributions of the haemodynamic power load are critical for characterizing/comparing disease states and for predicting the performance of cardiovascular devices. However, haemodynamic performance parameters vary significantly from subject to subject because of body size, heart rate and subject-specific CO. Therefore, a 'normalized' energy dissipation index, as a function of the 'non-dimensional' physical parameters that govern the circulation networks, is needed for comparative/integrative biological studies and clinical decision-making. In this paper, a complete network-independent non-dimensional formulation that incorporates pulsatile flow regimes is developed. Mechanical design variables of cardiovascular flow systems are identified and the Buckingham Pi theorem is formally applied to obtain the corresponding non-dimensional scaling parameter sets. Two scaling approaches are considered to address both the lumped parameter networks and the distributed circulation components. The validity of these non-dimensional number sets is tested extensively through the existing empirical allometric scaling laws of circulation systems. Additional validation studies are performed using a parametric numerical arterial model that represents the transmission and windkessel characteristics, which are adjusted to represent different body sizes and non-dimensional haemodynamic states. Simulations demonstrate that the proposed non-dimensional indices are independent of body size for healthy conditions, but are sensitive to deviations caused by off-design disease states that alter the energetic load. Sensitivity simulations are used to identify the relationship between pulsatile power loss and non-dimensional characteristics, and optimal operational states are computed.