Pressure-Poisson equation in numerical simulation of cerebral arterial circulation and its effect on the electrical conductivity of the brain

BACKGROUND AND OBJECTIVE

This study considers dynamic modeling of the cerebral arterial circulation and reconstructing an atlas for the electrical conductivity of the brain. Electrical conductivity is a governing parameter in several electrophysiological modalities applied in neuroscience, such as electroencephalography (EEG), transcranial electrical stimulation (tES), and electrical impedance tomography (EIT). While high-resolution 7-Tesla (T) Magnetic Resonance Imaging (MRI) data allow for reconstructing the cerebral arteries with a cross-sectional diameter larger than the voxel size, electrical conductivity cannot be directly inferred from MRI data. Brain models of electrophysiology typically associate each brain tissue compartment with a constant electrical conductivity, omitting any dynamic effects of cerebral blood circulation. Incorporating those effects poses the challenge of solving a system of incompressible Navier-Stokes equations (NSEs) in a realistic multi-compartment head model. However, using a simplified circulation model is well-motivated since, on the one hand, the complete system does not always have a numerically stable solution and, on the other hand, the full set of arteries cannot be perfectly reconstructed from the MRI data, meaning that any solution will be approximative.

METHODS

We postulate that circulation in the distinguishable arteries can be estimated via the pressure-Poisson equation (PPE), which is coupled with Fick's law of diffusion for microcirculation. To establish a fluid exchange model between arteries and microarteries, a boundary condition derived from the Hagen-Poisseuille model is applied. The relationship between the estimated volumetric blood concentration and the electrical conductivity of the brain tissue is approximated through Archie's law for fluid flow in porous media.

RESULTS

Through the formulation of the PPE and a set of boundary conditions (BCs) based on the Hagen-Poisseuille model, we obtained an equivalent formulation of the incompressible Stokes equation (SE). Thus, allowing effective blood pressure estimation in cerebral arteries segmented from open 7T MRI data.

CONCLUSIONS

As a result of this research, we developed and built a useful modeling framework that accounts for the effects of dynamic blood flow on a novel MRI-based electrical conductivity atlas. The electrical conductivity perturbation obtained in numerical experiments has an appropriate overall match with previous studies on this subject. Further research to validate these results will be necessary.

Direct 0D-3D coupling of a lattice Boltzmann methodology for fluid-structure aortic flow simulations

This work introduces a numerical approach and implementation for the direct coupling of arbitrary complex ordinary differential equation- (ODE-)governed zero-dimensional (0D) boundary conditions to three-dimensional (3D) lattice Boltzmann-based fluid-structure systems for hemodynamics studies. In particular, a most complex configuration is treated by considering a dynamic left ventricle- (LV-)elastance heart model which is governed by (and applied as) a nonlinear, non-stationary hybrid ODE-Dirichlet system. Other ODE-based boundary conditions, such as lumped parameter Windkessel models for truncated vasculature, are also considered. Performance studies of the complete 0D-3D solver, including its treatment of the lattice Boltzmann fluid equations and elastodynamics equations as well as their interactions, is conducted through a variety of benchmark and convergence studies that demonstrate the ability of the coupled 0D-3D methodology in generating physiological pressure and flow waveforms-ultimately enabling the exploration of various physical and physiological parameters for hemodynamics studies of the coupled LV-arterial system. The methods proposed in this paper can be easily applied to other ODE-based boundary conditions as well as to other fluid problems that are modeled by 3D lattice Boltzmann equations and that require direct coupling of dynamic 0D boundary conditions.

Applied machine learning for blood pressure estimation using a small, real-world electrocardiogram and photoplethysmogram dataset

Applying machine learning techniques to electrocardiography and photoplethysmography signals and their multivariate-derived waveforms is an ongoing effort to estimate non-occlusive blood pressure. Unfortunately, real ambulatory electrocardiography and photoplethysmography waveforms are inevitably affected by motion and noise artifacts, so established machine learning architectures perform poorly when trained on data of the Multiparameter Intelligent Monitoring in Intensive Care II type, a publicly available ICU database. Our study addresses this problem by applying four well-established machine learning methods, i.e., random forest regression, support vector regression, Adaboost regression and artificial neural networks, to a small, self-sampled electrocardiography-photoplethysmography dataset (n = 54) to improve the robustness of machine learning to real-world BP estimates. We evaluated the performance using a selection of optimal feature morphologies of waveforms by using pulse arrival time, morphological and frequency photoplethysmography parameters and heart rate variability as characterization data. On the basis of the root mean square error and mean absolute error, our study showed that support vector regression gave the best performance for blood pressure estimation from noisy data, achieving an mean absolute error of 6.97 mmHg, which meets the level C criteria set by the British Hypertension Society. We demonstrate that ambulatory electrocardiography- photoplethysmography signals acquired by mobile discrete devices can be used to estimate blood pressure.

Sensitivity analysis and uncertainty quantification of 1-D models of pulmonary hemodynamics in mice under control and hypertensive conditions

Pulmonary hypertension (PH), defined as an elevated mean blood pressure in the main pulmonary artery (MPA) at rest, is associated with vascular remodeling of both large and small arteries. PH has several sub-types that are all linked to high mortality rates. In this study, we use a one-dimensional (1-D) fluid dynamics model driven by in vivo measurements of MPA flow to understand how model parameters and network size influence MPA pressure predictions in the presence of PH. We compare model predictions with in vivo MPA pressure measurements from a control and a hypertensive mouse and analyze results in three networks of increasing complexity, extracted from micro-computed tomography (micro-CT) images. We introduce global scaling factors for boundary condition parameters and perform local and global sensitivity analysis to calculate parameter influence on model predictions of MPA pressure and correlation analysis to determine a subset of identifiable parameters. These are inferred using frequentist optimization and Bayesian inference via the Delayed Rejection Adaptive Metropolis (DRAM) algorithm. Frequentist and Bayesian uncertainty is computed for model parameters and MPA pressure predictions. Results show that MPA pressure predictions are most sensitive to distal vascular resistance and that parameter influence changes with increasing network complexity. Our outcomes suggest that PH leads to increased vascular stiffness and decreased peripheral compliance, congruent with clinical observations.

Personalized 0D models of normal and stenosed carotid arteries

BACKGROUND AND OBJECTIVE

Recent advances in medical imaging like MRI, CT-Scan, Doppler ultrasound, etc. have made it possible to study the hemodynamics of cardiovascular system having different levels of vessel abnormalities.

METHODS

Within this work, we have developed two different personalized lumped-parameter models of the human carotid arteries having elastic and viscoelastic vessel wall behaviors. The data used in developing the models of the carotid arteries is taken from a healthy subject and a patient having mild carotid stenosis (55%) near a bifurcation using doppler ultrasound. The data consists measurements of blood flow velocities and geometrical parameters at selected locations. Prior to the measurements, the key measurable geometrical parameters are identified by normalized local sensitivity analysis.

RESULTS

Finally, both developed and personalized models of carotid arteries are validated against the blood flow measurements obtained near carotid bifurcation. We observe a good agreement between model simulations and blood flow measurements taken near the bifurcation i.e. (r=0.94) for the healthy subject and (r=0.96) for the patient having a stenosis near the bifurcation.

CONCLUSIONS

This work provides further evidence, that the hemodynamics near a bifurcation can be modelled well with a 0D approach, even with different levels of stenosis.

Effect of Viscoelasticity on Arterial-Like Pulsatile Flow Dynamics and Energy

Time-dependent arterial wall property is an important but difficult topic in vascular mechanics. Hysteresis, which appears during the measurement of arterial pressure-diameter relationship through a cardiac cycle, has been used to indicate time-dependent mechanics of arteries. However, the cause-effect relationship between viscoelastic (VE) properties of the arterial wall and hemodynamics, particularly the viscous contribution to hemodynamics, remains challenging. Herein, we show direct comparisons between elastic (E) (loss/storage < 0.1) and highly viscoelastic (loss/storage > 0.45) conduit structures with arterial-like compliance, in terms of their capability of altering pulsatile flow, wall shear, and energy level. Conduits were made from varying ratio of vinyl- and methyl-terminated poly(dimethylsiloxane) and were fit in a mimetic circulatory system measuring volumetric flow, pressure, and strain. Results indicated that when compared to elastic conduits, viscoelastic conduits attenuated lumen distension waveforms, producing an average of 11% greater cross-sectional area throughout a mimetic cardiac cycle. In response to such changes in lumen diameter strain, pressure and volumetric flow waves in viscoelastic conduits decreased by 3.9% and 6%, respectively, in the peak-to-peak amplitude. Importantly, the pulsatile waveforms for both diameter strain and volumetric flow demonstrated greater temporal alignment in viscoelastic conduits due to pulsation attenuation, resulting in 25% decrease in the oscillation of wall shear stress (WSS). We hope these findings may be used to further examine time-dependent arterial properties in disease prognosis and progression, as well as their use in vascular graft design.

On the accuracy of displacement-based wave intensity analysis: Effect of vessel wall viscoelasticity and nonlinearity

Recent studies showed that wave intensity analysis (WIA) provides clinically valuable information about local and global cardiovascular function. Wave intensity (WI) is computed as the product of the pressure change and the velocity change during short time intervals. The major limitation of WIA in clinical practice is the need for invasive pressure measurement. Since vessel wall displacement can be measured non-invasively, the usage of WI will be expanded if the vessel wall dilation is used instead of pressure in derivation of WI waveform. Our goal in this study is to investigate the agreement between wall displacement-based WI and the pressure-based WI for different vessel wall models including linear elastic, nonlinear and viscoelastic cases. The arbitrary Eulerian Lagrangian finite element method is employed to solve the coupled fluid-structure interaction (FSI). Our computational models also include two types of vascular disease-related cases with geometrical irregularities, aneurysm and stenosis. Our results show that for vessels with linear elastic wall, the displacement-based WI is almost identical to the pressure-based WI. The existence of vessel irregularities does not impact the accuracy of displacement-based WI. However, in a viscoelastic wall where there is a phase difference between pressure and vessel wall dilation, displacement-based WI deviated from pressure-based WI. The error associated with this phase difference increased nonlinearly with increasing viscosity. This results in a maximum error of 6.8% and 7.13% for a regular viscoelastic vessel wall and an irregular viscoelastic vessel wall, respectively. A separate analysis has also been performed on the agreement of backward and forward running waves extracted from a decomposition of the displacement-based and pressure-based WI. Our findings suggest that displacement-based WI is a reliable method of WIA for large central arteries that do not show viscoelastic behaviors. This can be clinically significant since the required information can be measured non-invasively.

An In-Vitro Flow Study Using an Artificial Circle of Willis Model for Validation of an Existing One-Dimensional Numerical Model

A one-dimensional (1D) numerical model has been previously developed to investigate the hemodynamics of blood flow in the entire human vascular network. In the current work, an experimental study of water-glycerin mixture flow in a 3D-printed silicone model of an anatomically accurate, complete circle of Willis (CoW) was conducted to investigate the flow characteristics in comparison with the simulated results by the 1D numerical model. In the experiment, the transient flow and pressure waveforms were measured at 13 selected segments within the flow network for comparisons. In the 1D simulation, the initial parameters of the vessel network were obtained by a direct measurement of the tubes in the experimental setup. The results verified that the 1D numerical model is able to capture the main features of the experimental pressure and flow waveforms with good reliability. The mean flow rates measurement results agree with the predictions of the 1D model with an overall difference of less than 1%. Further experiment might be needed to validate the 1D model in capturing pressure waveforms.

Proinflammatory Arterial Stiffness Syndrome: A Signature of Large Arterial Aging

Age-associated structural and functional remodeling of the arterial wall produces a productive environment for the initiation and progression of hypertension and atherosclerosis. Chronic aging stress induces low-grade proinflammatory signaling and causes cellular proinflammation in arterial walls, which triggers the structural phenotypic shifts characterized by endothelial dysfunction, diffuse intimal-medial thickening, and arterial stiffening. Microscopically, aged arteries exhibit an increase in arterial cell senescence, proliferation, invasion, matrix deposition, elastin fragmentation, calcification, and amyloidosis. These characteristic cellular and matrix alterations not only develop with aging but can also be induced in young animals under experimental proinflammatory stimulation. Interestingly, these changes can also be attenuated in old animals by reducing low-grade inflammatory signaling. Thus, mitigating age-associated proinflammation and arterial phenotype shifts is a potential approach to retard arterial aging and prevent the epidemic of hypertension and atherosclerosis in the elderly.

Uncertainty Quantification in a Patient-Specific One-Dimensional Arterial Network Model: EnKF-Based Inflow Estimator

Successful clinical use of patient-specific models for cardiovascular dynamics depends on the reliability of the model output in the presence of input uncertainties. For 1D fluid dynamics models of arterial networks, input uncertainties associated with the model output are related to the specification of vessel and network geometry, parameters within the fluid and wall equations, and parameters used to specify inlet and outlet boundary conditions. This study investigates how uncertainty in the flow profile applied at the inlet boundary of a 1D model affects area and pressure predictions at the center of a single vessel. More specifically, this study develops an iterative scheme based on the ensemble Kalman filter (EnKF) to estimate the temporal inflow profile from a prior distribution of curves. The EnKF-based inflow estimator provides a measure of uncertainty in the size and shape of the estimated inflow, which is propagated through the model to determine the corresponding uncertainty in model predictions of area and pressure. Model predictions are compared to ex vivo area and blood pressure measurements in the ascending aorta, the carotid artery, and the femoral artery of a healthy male Merino sheep. Results discuss dynamics obtained using a linear and a nonlinear viscoelastic wall model.

Linear and Nonlinear Viscoelastic Arterial Wall Models: Application on Animals

This work deals with the viscoelasticity of the arterial wall and its influence on the pulse waves. We describe the viscoelasticity by a nonlinear Kelvin–Voigt model in which the coefficients are fitted using experimental time series of pressure and radius measured on a sheep's arterial network. We obtained a good agreement between the results of the nonlinear Kelvin–Voigt model and the experimental measurements. We found that the viscoelastic relaxation time—defined by the ratio between the viscoelastic coefficient and the Young's modulus—is nearly constant throughout the network. Therefore, as it is well known that smaller arteries are stiffer, the viscoelastic coefficient rises when approaching the peripheral sites to compensate the rise of the Young's modulus, resulting in a higher damping effect. We incorporated the fitted viscoelastic coefficients in a nonlinear 1D fluid model to compute the pulse waves in the network. The damping effect of viscoelasticity on the high-frequency waves is clear especially at the peripheral sites.

WAVE PROPAGATION IN A 1D FLUID DYNAMICS MODEL USING PRESSURE-AREA MEASUREMENTS FROM OVINE ARTERIES

This study considers a 1D fluid dynamics arterial network model with 14 vessels developed to assimilate ex vivo 0D temporal data for pressure-area dynamics in individual vessel segments from 11 male Merino sheep. A 0D model was used to estimate vessel wall parameters in a two-parameter elastic model and a four-parameter Kelvin viscoelastic model. This was done using nonlinear optimization minimizing the least squares error between model predictions and measured cross-sectional areas. Subsequently, estimated values for elastic stiffness and unstressed area were related to construct a nonlinear relationship. This relation was used in the network model. A 1D single vessel model of the aorta was then developed and used to estimate the inflow profile and parameters for total resistance and compliance for the downstream network and to demonstrate effects of incorporating viscoelasticity in the arterial wall. Lastly, the extent to which vessel wall parameters estimated from ex vivo data can be used to realistically simulate pressure and area in a vessel network was evaluated. Elastic wall parameters in the network simulations were found to yield pressure-area relationships across all vessel locations and sheep that were in ranges comparable to those in the ex vivo data.

Fluid friction and wall viscosity of the 1D blood flow model

We study the behavior of the pulse waves of water into a flexible tube for application to blood flow simulations. In pulse waves both fluid friction and wall viscosity are damping factors, and difficult to evaluate separately. In this paper, the coefficients of fluid friction and wall viscosity are estimated by fitting a nonlinear 1D flow model to experimental data. In the experimental setup, a distensible tube is connected to a piston pump at one end and closed at another end. The pressure and wall displacements are measured simultaneously. A good agreement between model predictions and experiments was achieved. For amplitude decrease, the effect of wall viscosity on the pulse wave has been shown as important as that of fluid viscosity.

A novel coupling algorithm for computing blood flow in viscoelastic arterial models

We propose a novel coupling algorithm, based on the operator-splitting scheme, which implements the viscoelastic wall law at the coupling nodes of the vessels. Two different viscoelastic models are used (V1 and V2), leading to five different computational setups: elastic wall law, model V1 applied at interior and coupling grid points, model V1 applied only at the interior grid points (V1-int), model V2 applied at interior and coupling grid points, model V2 applied only at the interior grid points (V2-int). These have been tested with two arterial configurations: (i) single artery, and (ii) complete arterial tree. Models V1-int and V2-int lead to incorrect conclusions and to errors which can be of the same order as, and are at least 1/5 of, the difference between the results with the elastic and the viscoelastic laws. Both test cases demonstrate the importance of modeling the viscous component of the pressure-area relationship at all grid points, including the coupling points between vessels or at the inlet/outlet of the model.

A viscoelastic fluid-structure interaction model for carotid arteries under pulsatile flow

In this study, a fluid-structure interaction model (FSI) incorporating viscoelastic wall behaviour is developed and applied to an idealized model of the carotid artery under pulsatile flow. The shear and bulk moduli of the arterial wall are described by Prony series, where the parameters can be derived from in vivo measurements. The aim is to develop a fully coupled FSI model that can be applied to realistic arterial geometries with normal or pathological viscoelastic wall behaviour. Comparisons between the numerical and analytical solutions for wall displacements demonstrate that the coupled model is capable of predicting the viscoelastic behaviour of carotid arteries. Comparisons are also made between the solid only and FSI viscoelastic models, and the results suggest that the difference in radial displacement between the two models is negligible.

Modelling and subject-specific validation of the heart-arterial tree system

A modeling approach integrated with a novel subject-specific characterization is here proposed for the assessment of hemodynamic values of the arterial tree. A 1D model is adopted to characterize large-to-medium arteries, while the left ventricle, aortic valve and distal micro-circulation sectors are described by lumped submodels. A new velocity profile and a new formulation of the non-linear viscoelastic constitutive relation suitable for the {Q, A} modeling are also proposed. The model is firstly verified semi-quantitatively against literature data. A simple but effective procedure for obtaining subject-specific model characterization from non-invasive measurements is then designed. A detailed subject-specific validation against in vivo measurements from a population of six healthy young men is also performed. Several key quantities of heart dynamics-mean ejected flow, ejection fraction, and left-ventricular end-diastolic, end-systolic and stroke volumes-and the pressure waveforms (at the central, radial, brachial, femoral, and posterior tibial sites) are compared with measured data. Mean errors around 5 and 8%, obtained for the heart and arterial quantities, respectively, testify the effectiveness of the model and its subject-specific characterization.

In vivo determination of elastic properties of the human aorta based on 4D ultrasound data

Computational analysis of the biomechanics of the vascular system aims at a better understanding of its physiology and pathophysiology. To be of clinical use, however, these models and thus their predictions, have to be patient specific regarding geometry, boundary conditions and material. In this paper we present an approach to determine individual material properties of human aortae based on a new type of in vivo full field displacement data acquired by dimensional time resolved three dimensional ultrasound (4D-US) imaging. We developed a nested iterative Finite Element Updating method to solve two coupled inverse problems: The prestrains that are present in the imaged diastolic configuration of the aortic wall are determined. The solution of this problem is integrated in an iterative method to identify the nonlinear hyperelastic anisotropic material response of the aorta to physiologic deformation states. The method was applied to 4D-US data sets of the abdominal aorta of five healthy volunteers and verified by a numerical experiment. This non-invasive in vivo technique can be regarded as a first step to determine patient individual material properties of the human aorta.

Experimental validation of a pulse wave propagation model for predicting hemodynamics after vascular access surgery

Hemodialysis patients require a vascular access that is, preferably, surgically created by connecting an artery and vein in the arm, i.e. an arteriovenous fistula (AVF). The site for AVF creation is chosen by the surgeon based on preoperative diagnostics, but AVFs are still compromised by flow-associated complications. Previously, it was shown that a computational 1D-model is able to describe pressure and flow after AVF surgery. However, predicted flows differed from measurements in 4/10 patients. Differences can be attributed to inaccuracies in Doppler measurements and input data, to neglecting physiological mechanisms or to an incomplete physical description of the pulse wave propagation after AVF surgery. The physical description can be checked by validating against an experimental setup consisting of silicone tubes mimicking the aorta and arm vasculature both before and after AVF surgery, which is the aim of the current study. In such an analysis, the output uncertainty resulting from measurement uncertainty in model input should be quantified. The computational model was fed by geometrical and mechanical properties collected from the setup. Pressure and flow waveforms were simulated and compared with experimental waveforms. The precision of the simulations was determined by performing a Monte Carlo study. It was concluded that the computational model was able to simulate mean pressures and flows accurately, whereas simulated waveforms were less attenuated than experimental ones, likely resulting from neglecting viscoelasticity. Furthermore, it was found that in the analysis output uncertainties, resulting from input uncertainties, cannot be neglected and should thus be considered.