Validation of a patient-specific one-dimensional model of the systemic arterial tree

The aim of this study is to develop and validate a patient-specific distributed model of the systemic arterial tree. This model is built using geometric and hemodynamic data measured on a specific person and validated with noninvasive measurements of flow and pressure on the same person, providing thus a patient-specific model and validation. The systemic arterial tree geometry was obtained from MR angiographic measurements. A nonlinear viscoelastic constitutive law for the arterial wall is considered. Arterial wall distensibility is based on literature data and adapted to match the wave propagation velocity of the main arteries of the specific subject, which were estimated by pressure waves traveling time. The intimal shear stress is modeled using the Witzig-Womersley theory. Blood pressure is measured using applanation tonometry and flow rate using transcranial ultrasound and phase-contrast-MRI. The model predicts pressure and flow waveforms in good qualitative and quantitative agreement with the in vivo measurements, in terms of wave shape and specific wave features. Comparison with a generic one-dimensional model shows that the patient-specific model better predicts pressure and flow at specific arterial sites. These results obtained let us conclude that a patient-specific one-dimensional model of the arterial tree is able to predict well pressure and flow waveforms in the main systemic circulation, whereas this is not always the case for a generic one-dimensional model.     

Uncertainty quantification in cerebral circulation simulations focusing on the collateral flow: Surrogate model approach with machine learning

Collateral circulation in the circle of Willis (CoW), closely associated with disease mechanisms and treatment outcomes, can be effectively investigated using one-dimensional–zero-dimensional hemodynamic simulations. As the entire cardiovascular system is considered in the simulation, it captures the systemic effects of local arterial changes, thus reproducing collateral circulation that reflects biological phenomena. The simulation facilitates rapid assessment of clinically relevant hemodynamic quantities under patient-specific conditions by incorporating clinical data. During patient-specific simulations, the impact of clinical data uncertainty on the simulated quantities should be quantified to obtain reliable results. However, as uncertainty quantification (UQ) is time-consuming and computationally expensive, its implementation in time-sensitive clinical applications is considered impractical. Therefore, we constructed a surrogate model based on machine learning using simulation data. The model accurately predicts the flow rate and pressure in the CoW in a few milliseconds. This reduced computation time enables the UQ execution with 100 000 predictions in a few minutes on a single CPU core and in less than a minute on a GPU. We performed UQ to predict the risk of cerebral hyperperfusion (CH), a life-threatening condition that can occur after carotid artery stenosis surgery if collateral circulation fails to function appropriately. We predicted the statistics of the postoperative flow rate increase in the CoW, which is a measure of CH, considering the uncertainties of arterial diameters, stenosis parameters, and flow rates measured using the patients’ clinical data. A sensitivity analysis was performed to clarify the impact of each uncertain parameter on the flow rate increase. Results indicated that CH occurred when two conditions were satisfied simultaneously: severe stenosis and when arteries of small diameter serve as the collateral pathway to the cerebral artery on the stenosis side. These findings elucidate the biological aspects of cerebral circulation in terms of the relationship between collateral flow and CH.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels

Time-domain-based one-dimensional wave propagation models of the arterial system are preferable over one-dimensional wave propagation models in the frequency domain since the latter neglect the non-linear convection forces present in the physiological situation, especially when the vessel is tapered. Moreover, one-dimensional wave propagation models of the arterial system can be used to provide boundary conditions for fully three-dimensional fluid-structure interaction computations that are usually defined in the time domain. In this study, a time-domain-based one-dimensional wave propagation model in a cross-sectional area, flow and pressure (A,q,p)-formulation is developed. Using this formulation, a constitutive law that includes viscoelasticity based on the mechanical behaviour of a Kelvin body, is introduced. The resulting pressure and flow waves travelling through a straight and tapered vessel are compared to experimental data obtained from measurements in an in vitro setup. The model presented shows to be well suited to predict wave propagation through these straight and tapered vessels with viscoelastic wall properties and hereto can serve as a time-domain-based method to model wave propagation in the human arterial system.     

Computational vascular fluid–structure interaction: methodology and application to cerebral aneurysms

A computational vascular fluid–structure interaction framework for the simulation of patient-specific cerebral aneurysm configurations is presented. A new approach for the computation of the blood vessel tissue prestress is also described. Simulations of four patient-specific models are carried out, and quantities of hemodynamic interest such as wall shear stress and wall tension are studied to examine the relevance of fluid–structure interaction modeling when compared to the rigid arterial wall assumption. We demonstrate that flexible wall modeling plays an important role in accurate prediction of patient-specific hemodynamics. Discussion of the clinical relevance of our methods and results is provided.     

On the choice of outlet boundary conditions for patient-specific analysis of aortic flow using computational fluid dynamics

Boundary conditions (BCs) are an essential part in computational fluid dynamics (CFD) simulations of blood flow in large arteries. Although several studies have investigated the influence of BCs on predicted flow patterns and hemodynamic wall parameters in various arterial models, there is a lack of comprehensive assessment of outlet BCs for patient-specific analysis of aortic flow. In this study, five different sets of outlet BCs were tested and compared using a subject-specific model of a normal aorta. Phase-contrast magnetic resonance imaging (PC-MRI) was performed on the same subject and velocity profiles extracted from the in vivo measurements were used as the inlet boundary condition. Computational results obtained with different outlet BCs were assessed in terms of their agreement with the PC-MRI velocity data and key hemodynamic parameters, such as pressure and flow waveforms and wall shear stress related indices. Our results showed that the best overall performance was achieved by using a well-tuned three-element Windkessel model at all model outlets, which not only gave a good agreement with in vivo flow data, but also produced physiological pressure waveforms and values. On the other hand, opening outlet BCs with zero pressure at multiple outlets failed to reproduce any physiologically relevant flow and pressure features.     

Inlet boundary conditions for blood flow simulations in truncated arterial networks

In the context of patient-specific cardiovascular applications, hemodynamics models (going from 3D to 0D) are often limited to a part of the arterial tree. This restriction implies the set up of artificial interfaces with the remaining parts of the cardiovascular system. In particular, the inlet boundary condition is crucial: it supplies the impulsion to the system and receives the reflected backward waves created by the distal network. Some aspects of this boundary condition need to be properly defined such as the treatment of backward waves (reflected or absorbed) and the value of the imposed hemodynamic wave (total or forward component). Most authors prescribe as inlet boundary condition (BC) the total measured variable (pressure, velocity or flow rate) in a reflective way. We show that with this type of inlet boundary condition, the model does not produce physiological waveforms. We suggest instead to prescribe only the forward component of the prescribed variable in an absorbing way. In this way, the computed reflected waves superpose with the prescribed forward waves to produce the total wave at the inlet. In this work, different inlet boundary conditions are implemented and compared for a 1D blood flow model. We test our boundary conditions on a truncated arterial model presented in the literature as well as on a patient-specific lower-limb model of a femoral bypass. We show that with this new boundary condition, a much better fitting is observed on the shape and intensity of the simulated pressure and velocity waves.     

Uncertainty propagation of phase contrast-MRI derived inlet boundary conditions in computational hemodynamics models of thoracic aorta

This study investigates the impact that uncertainty in phase contrast-MRI derived inlet boundary conditions has on patient-specific computational hemodynamics models of the healthy human thoracic aorta. By means of Monte Carlo simulations, we provide advice on where, when and how, it is important to account for this source of uncertainty. The study shows that the uncertainty propagates not only to the intravascular flow, but also to the shear stress distribution at the vessel wall. More specifically, the results show an increase in the uncertainty of the predicted output variables, with respect to the input uncertainty, more marked for blood pressure and wall shear stress. The methodological approach proposed here can be easily extended to study uncertainty propagation in both healthy and pathological computational hemodynamic models.     

Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements

The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagation in Voigt-type visco-elastic arteries was tested against measurements in a well-defined experimental 1:1 replica of the 37 largest conduit arteries in the human systemic circulation. The parameters required by the numerical algorithm were directly measured in the in vitro setup and no data fitting was involved. The inclusion of wall visco-elasticity in the numerical model reduced the underdamped high-frequency oscillations obtained using a purely elastic tube law, especially in peripheral vessels, which was previously reported in this paper [Matthys et al., 2007. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. 40, 3476-3486]. In comparison to the purely elastic model, visco-elasticity significantly reduced the average relative root-mean-square errors between numerical and experimental waveforms over the 70 locations measured in the in vitro model: from 3.0% to 2.5% (p<0.012) for pressure and from 15.7% to 10.8% (p<0.002) for the flow rate. In the frequency domain, average relative errors between numerical and experimental amplitudes from the 5th to the 20th harmonic decreased from 0.7% to 0.5% (p<0.107) for pressure and from 7.0% to 3.3% (p<10(-6)) for the flow rate. These results provide additional support for the use of 1-D reduced modelling to accurately simulate clinically relevant problems at a reasonable computational cost.     

Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements

A numerical model based on the nonlinear, one-dimensional (1-D) equations of pressure and flow wave propagation in conduit arteries is tested against a well-defined experimental 1:1 replica of the human arterial tree. The tree consists of 37 silicone branches representing the largest central systemic arteries in the human, including the aorta, carotid arteries and arteries that perfuse the upper and lower limbs and the main abdominal organs. The set-up is mounted horizontally and connected to a pulsatile pump delivering a periodic output similar to the aortic flow. Terminal branches end in simple resistance models, consisting of stiff capillary tubes leading to an overflow reservoir that reflects a constant venous pressure. The parameters required by the numerical algorithm are directly measured in the in vitro set-up and no data fitting is involved. Comparison of experimental and numerical pressure and flow waveforms shows the ability of the 1-D time-domain formulation to capture the main features of pulse wave propagation measured throughout the system test. As a consequence of the simple resistive boundary conditions used to reduce the uncertainty of the parameters involved in the simulation, the experimental set-up generates waveforms at terminal branches with additional non-physiological oscillations. The frequencies of these oscillations are well captured by the 1-D model, even though amplitudes are overestimated. Adding energy losses in bifurcations and including fluid inertia and compliance to the purely resistive terminal models does not reduce the underdamped effect, suggesting that wall visco-elasticity might play an important role in the experimental results. Nevertheless, average relative root-mean-square errors between simulations and experimental waveforms are smaller than 4% for pressure and 19% for the flow at all 70 locations studied.     

Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow

It is well known that blood vessels exhibit viscoelastic properties, which are modeled in the literature with different mathematical forms and experimental bases. The wide range of existing viscoelastic wall models may produce significantly different blood flow, pressure, and vessel deformation solutions in cardiovascular simulations. In this paper, we present a novel comparative study of two different viscoelastic wall models in nonlinear one-dimensional (1D) simulations of blood flow. The viscoelastic models are from papers by Holenstein et al. in 1980 (model V1) and Valdez-Jasso et al. in 2009 (model V2). The static elastic or zero-frequency responses of both models are chosen to be identical. The nonlinear 1D blood flow equations incorporating wall viscoelasticity are solved using a space-time finite element method and the implementation is verified with the Method of Manufactured Solutions. Simulation results using models V1, V2 and the common static elastic model are compared in three application examples: (i) wave propagation study in an idealized vessel with reflection-free outflow boundary condition; (ii) carotid artery model with nonperiodic boundary conditions; and (iii) subject-specific abdominal aorta model under rest and simulated lower limb exercise conditions. In the wave propagation study the damping and wave speed were largest for model V2 and lowest for the elastic model. In the carotid and abdominal aorta studies the most significant differences between wall models were observed in the hysteresis (pressure-area) loops, which were larger for V2 than V1, indicating that V2 is a more dissipative model. The cross-sectional area oscillations over the cardiac cycle were smaller for the viscoelastic models compared to the elastic model. In the abdominal aorta study, differences between constitutive models were more pronounced under exercise conditions than at rest. Inlet pressure pulse for model V1 was larger than the pulse for V2 and the elastic model in the exercise case. In this paper, we have successfully implemented and verified two viscoelastic wall models in a nonlinear 1D finite element blood flow solver and analyzed differences between these models in various idealized and physiological simulations, including exercise. The computational model of blood flow presented here can be utilized in further studies of the cardiovascular system incorporating viscoelastic wall properties.     

On the propagation of a wave front in viscoelastic arteries

In formulating a mathematical model of the arterial system, the one-dimensional flow approximation yields realistic pressure and flow pulses in the proximal as well as in the distal regions of a simulated arterial conduit, provided that the viscoelastic damping induced by the vessel wall is properly taken into account. Models which are based on a purely elastic formulation of the arterial wall properties are known to produce shocklike transitions in the propagating pulses which are not observed in man under physiological conditions. The viscoelastic damping characteristics are such that they are expected to reduce the tendency of shock formation in the model. In order to analyze this phenomenon, the propagation of first and second-order pressure waves is calculated with the aid of a wave front expansion, and criteria for the formation of shocks are derived. The application of the results to the human arterial system show that shock waves are not to be expected under normal conditions, while in case of a pathologically increased pressure rise at the root of the aorta, shocklike transitions may develop in the periphery. In particular, it is shown that second-order waves never lead to shock formation in finite time for the class of initial conditions and mechanical wave guides which are of interest in the mammalian circulation.     

Identification of vascular territory resistances in one-dimensional hemodynamics simulations

The present work deals with the parameter identification problem in outflow models used in one-dimensional simulations of arterial blood flow. Specifically, the resistive elements that define the models used to account for the blood supply to the vascular territories in arterial networks are computed by solving a system of non-linear equations using a Broyden method. This strategy is employed to compute the terminal parameters in the vascular territories of an anatomically detailed model of the arm comprising 67 arterial segments and 16 vascular territories. A comparison with a simple analytical approach, in terms of vascular territory resistances, average blood flows and time-dependent hemodynamic quantities, is performed. Also, a sensitivity analysis is presented to assess the performance of this new approach in normal and abnormal cardiovascular scenarios. This identification procedure allows to correctly set up hemodynamics simulations in highly detailed arterial networks making possible to gain insight in the aspects related to the blood circulation in arterial vessels.     

A coupled hydrodynamic model of the cardiovascular and cerebrospinal fluid system

Coupling of the cardiovascular and cerebrospinal fluid (CSF) system is considered to be important to understand the pathophysiology of cerebrovascular and craniospinal disease and intrathecal drug delivery. A coupled cardiovascular and CSF system model was designed to examine the relation of spinal cord (SC) blood flow (SCBF) and CSF pulsations along the spinal subarachnoid space (SSS). A one-dimensional (1-D) cardiovascular tree model was constructed including a simplified SC arterial network. Connection between the cardiovascular and CSF system was accomplished by a transfer function based on in vivo measurements of CSF and cerebral blood flow. A 1-D tube model of the SSS was constructed based on in vivo measurements in the literature. Pressure and flow throughout the cardiovascular and CSF system were determined for different values of craniospinal compliance. SCBF results indicated that the cervical, thoracic, and lumbar SC each had a signature waveform shape. The cerebral blood flow to CSF transfer function reproduced an in vivo-like CSF flow waveform. The 1-D tube model of the SSS resulted in a distribution of CSF pressure and flow and a wave speed that were similar to those in vivo. The SCBF to CSF pulse delay was found to vary a great degree along the spine depending on craniospinal compliance and vascular anatomy. The properties and anatomy of the SC arterial network and SSS were found to have an important impact on pressure and flow and perivascular fluid movement to the SC. Overall, the coupled model provides predictions about the flow and pressure environment in the SC and SSS. More detailed measurements are needed to fully validate the model.     

Three-dimensional hemodynamics analysis of the circle of Willis in the patient-specific nonintegral arterial structures

The hemodynamic alteration in the cerebral circulation caused by the geometric variations in the cerebral circulation arterial network of the circle of Wills (CoW) can lead to fatal ischemic attacks in the brain. The geometric variations due to impairment in the arterial network result in incomplete cerebral arterial structure of CoW and inadequate blood supply to the brain. Therefore, it is of great importance to understand the hemodynamics of the CoW, for efficiently and precisely evaluating the status of blood supply to the brain. In this paper, three-dimensional computational fluid dynamics of the main CoW vasculature coupled with zero-dimensional lumped parameter model boundary condition for the CoW outflow boundaries is developed for analysis of the blood flow distribution in the incomplete CoW cerebral arterial structures. The geometric models in our study cover the arterial segments from the aorta to the cerebral arteries, which can allow us to take into account the innate patient-specific resistance of the arterial trees. Numerical simulations of the governing fluid mechanics are performed to determine the CoW arterial structural hemodynamics, for illustrating the redistribution of the blood flow in CoW due to the structural variations. We have evaluated our coupling methodology in five patient-specific cases that were diagnosed with the absence of efferent vessels or impairment in the connective arteries in their CoWs. The velocity profiles calculated by our approach in the segments of the patient-specific arterial structures are found to be very close to the Doppler ultrasound measurements. The accuracy and consistency of our hemodynamic results have been improved (to \(16.1 \pm 18.5\) %) compared to that of the pure-resistance boundary conditions (of 43.5 \(\pm \) 28 %). Based on our grouping of the five cases according to the occurrence of unilateral occlusion in vertebral arteries, the inter-comparison has shown that (i) the flow reduction in posterior cerebral arteries is the consequence of the unilateral vertebral arterial occlusion, and (ii) the flow rate in the anterior cerebral arteries is correlated with the posterior structural variations. This study shows that our coupling approach is capable of providing comprehensive information of the hemodynamic alterations in the pathological CoW arterial structures. The information generated by our methodology can enable evaluation of both the functional and structural status of the clinically significant symptoms, for assisting the treatment decision-making.     

Probabilistic noninvasive prediction of wall properties of abdominal aortic aneurysms using Bayesian regression

Multiple patient-specific parameters, such as wall thickness, wall strength, and constitutive properties, are required for the computational assessment of abdominal aortic aneurysm (AAA) rupture risk. Unfortunately, many of these quantities are not easily accessible and could only be determined by invasive procedures, rendering a computational rupture risk assessment obsolete. This study investigates two different approaches to predict these quantities using regression models in combination with a multitude of noninvasively accessible, explanatory variables. We have gathered a large dataset comprising tensile tests performed with AAA specimens and supplementary patient information based on blood analysis, the patients medical history, and geometric features of the AAAs. Using this unique database, we harness the capability of state-of-the-art Bayesian regression techniques to infer probabilistic models for multiple quantities of interest. After a brief presentation of our experimental results, we show that we can effectively reduce the predictive uncertainty in the assessment of several patient-specific parameters, most importantly in thickness and failure strength of the AAA wall. Thereby, the more elaborate Bayesian regression approach based on Gaussian processes consistently outperforms standard linear regression. Moreover, our study contains a comparison to a previously proposed model for the wall strength.     

Image-based modeling of hemodynamics in coronary artery aneurysms caused by Kawasaki disease

Kawasaki Disease (KD) is the leading cause of acquired pediatric heart disease. A subset of KD patients develops aneurysms in the coronary arteries, leading to increased risk of thrombosis and myocardial infarction. Currently, there are limited clinical data to guide the management of these patients, and the hemodynamic effects of these aneurysms are unknown. We applied patient-specific modeling to systematically quantify hemodynamics and wall shear stress in coronary arteries with aneurysms caused by KD. We modeled the hemodynamics in the aneurysms using anatomic data obtained by multi-detector computed tomography (CT) in a 10-year-old male subject who suffered KD at age 3?years. The altered hemodynamics were compared to that of a reconstructed normal coronary anatomy using our subject as the model. Computer simulations using a robust finite element framework were used to quantify time-varying shear stresses and particle trajectories in the coronary arteries. We accounted for the cardiac contractility and the microcirculation using physiologic downstream boundary conditions. The presence of aneurysms in the proximal coronary artery leads to flow recirculation, reduced wall shear stress within the aneurysm, and high wall shear stress gradients at the neck of the aneurysm. The wall shear stress in the KD subject (2.95-3.81 dynes/sq cm) was an order of magnitude lower than the normal control model (17.10-27.15 dynes/sq cm). Particle residence times were significantly higher, taking 5 cardiac cycles to fully clear from the aneurysmal regions in the KD subject compared to only 1.3 cardiac cycles from the corresponding regions of the normal model. In this novel quantitative study of hemodynamics in coronary aneurysms caused by KD, we documented markedly abnormal flow patterns that are associated with increased risk of thrombosis. This methodology has the potential to provide further insights into the effects of aneurysms in KD and to help risk stratify patients for appropriate medical and surgical interventions.     

Coupled computational analysis of arterial LDL transport -- effects of hypertension

Hypertension, a risk factor for atherosclerosis, increases the uptake of low density lipoproteins (LDL) by the arterial wall. Our objective in this work was to use computational modeling to identify physical factors that could be partially responsible for this effect. Fluid flow and mass transfer patterns in the lumen and wall of an arterial model were computed in a coupled manner, replicating as closely as possible previous experimental studies in which LDL uptake into the artery wall was measured in straight, excised arterial segments. Under conditions of both flow and no-flow, simulations predicted an increase in concentration polarization of LDL at the artery wall when arterial pressure was increased from 120 to 160 mmHg. However, this led to only a slight increase in mean LDL concentration within the arterial wall. However, if the permeability of the endothelium to LDL was allowed to vary with intra-arterial pressure, then the simulations predicted that the uptake of LDL would be enhanced 1.9-2.6 fold at higher pressure. The magnitude of this increase was consistent with experimental data. We conclude that the concentration polarization effects, enhanced by elevated intra-arterial pressure, cannot explain the increase in LDL uptake seen under hypertensive conditions. Instead, the data are most consistent with a pressure-linked increase in endothelial permeability to LDL.     

A multiscale approach for modelling wave propagation in an arterial segment

A mathematical model of blood flow through an arterial vessel is presented and the wave propagation in it is studied numerically. Based on the assumption of long wavelength and small amplitude of the pressure waves, a quasi-one-dimensional (1D) differential model is adopted. It describes the non-linear fluid-wall interaction and includes wall deformation in both radial and axial directions. The 1D model is coupled with a six compartment lumped parameter model, which accounts for the global circulatory features and provides boundary conditions. The differential equations are first linearized to investigate the nature of the propagation phenomena. The full non-linear equations are then approximated with a numerical finite difference method on a staggered grid. Some numerical simulations show the characteristics of the wave propagation. The dependence of the flow, of the wall deformation and of the wave velocity on the elasticity parameter has been highlighted. The importance of the axial deformation is evidenced by its variation in correspondence of the pressure peaks. The wave disturbances consequent to a local stiffening of the vessel and to a compliance jump due to prosthetic implantations are finally studied.