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.     

A finite-element model of blood flow in arteries including taper, branches, and obstructions

A nonlinear mathematical model of arterial blood flow, which can account for tapering, branching, and the presence of stenosed segments, is presented. With the finite-element method, the model equations are transformed into a system of algebraic equations that can be solved on a high-speed digital computer to yield values of pressure and volume rate of flow as functions of time and arterial position. A model of the human femoral artery is used to compare the effects of linear and nonlinear modeling. During periods of rapid alternations in pressure or flow, the nonlinear model shows significantly different results than the linear model. The effect of a stenosis on pressure and flow waveforms is also simulated, and the results indicate that these waveforms are significantly altered by moderate and severe stenoses.     

Computer modeling of the human systemic arterial tree

A model of the human systemic arterial tree has been devised, based on a lumped-parameter-circuit approximate form. This model has been set up and studied on an analog computer. A feature of this simulation is the division of the arterial system into sections whose lengths are inversely proportional (approximately) to their cross-sectional area-or what is termed ‘equal-volume’ modeling.

Great care was exercised in the determination of the model parameters, using expressions for these parameters from a recent paper by Rideout and Dick on fluid flow in distensible tubes, with numerical values based on measurements reported in the medical literature.

The simulated pressure and flow waveforms obtained with the model compare favorably with data recorded from the normal adult human, and exhibit such well-known features as distal delay and peaking of pressure pulses. The aortic input impedance vs. frequency curve checks well against measurements on the human. The model also provides a simple means for determination of cardiac output, cardiac work and cardiac power under various assumed conditions such as variation of heart rate.     

Open, reconfigurable cytometric acquisition system: ORCAS.

A digital signal processing (DSP)-based digital data acquisition system has been developed to support novel flow cytometry efforts. The system flexibility includes how it detects, captures, and processes event data. Custom data capture boards utilizing analog to digital converters (ADCs) and field programmable gate arrays (FPGA) detect events and capture correlated event data. A commercial DSP board processes the captured data and sends the results over the IEEE 1394 bus to the host computer that provides a user interface for acquisition, display, analysis, and storage. The system collects list mode data, correlated pulse shapes, or streaming data from a variety of detector types using Linux, Mac OS X, and Windows host computers. It extracts pulse features not found on commercial systems with excellent sensitivity and linearity over a wide dynamic range. List mode data are saved in FCS 3.0 formatted files while streaming or correlated waveform data are saved in custom format files for postprocessing. Open, reconfigurable cytometric acquisition system is compact, scaleable, flexible, and modular. Programmable feature extraction algorithms have exciting possibilities for both new and existing applications. The recent availability of a commercial data capture board will enable general availability of similar systems.     

An evaluation of some electrical waveforms and voltages used for electric fishing; with special reference to their use in backpack electric fishing gear

Exponential pulse waveform in electric fishing gave a significantly poorer capture efficiency than other waveforms evaluated. Plasma cortisol levels were significantly and uniformly elevated after shocking by all waveforms. Mortality from the electric fishing was low, 0·2% overall, with few sub-lethal effects. A 0·5 ms duration square pulse waveform had significantly higher catch-per-unit-power than the other waveforms.     

Modelling pulse wave propagation in the rabbit systemic circulation to assess the effects of altered nitric oxide synthesis

Pulse wave propagation in the mature rabbit systemic circulation was simulated using the one-dimensional equations of blood flow in compliant vessels. A corrosion cast of the rabbit circulation was manufactured to obtain arterial lengths and diameters. Pulse wave speeds and inflow and outflow boundary conditions were derived from in vivo data. Numerical results captured the main features of in vivo pressure and velocity pulse waveforms in the aorta, brachiocephalic artery and central ear artery. This model was used to elucidate haemodynamic mechanisms underlying changes in peripheral pulse waveforms observed in vivo after administering drugs that alter nitric oxide synthesis in the endothelial cells lining blood vessels. According to our model, these changes can be explained by single or combined alterations of blood viscosity, peripheral resistance and compliance, and the elasticity of conduit arteries.     

Noninvasive pressure pulse waveform analysis of flow-mediated vasodilation evoked by post-occlusive reactive hyperemia maneuver

Post-occlusive reactive hyperemia (PORH) assesses flow-mediated vasodilation at microvascular level due to bioactivity of endothelial-derived factors. Ordinary augmentation index that quantifies endothelial response is based on an ensemble-averaged waveform that limits its short-time application. This study proposes a mathematical model and two corresponding indices to evaluate arterial pressure response after blood flow restoration. Radial pressure pulse waveforms were acquired by a 12 bits acquisition board at a sampling rate of 1.0 kHz using a piezoelectric transducer. Signals were stored during 30 s at baseline condition and 60 s after 5-min occlusion using an arm-cuff placed over the brachial artery. In both conditions, the pressure pulse waveform presents systolic and diastolic phases with progressive and regressive pulse waveforms, respectively. Changes in pulse wave morphology were also observed and comprised attenuation of the pulse pressure amplitude (markedly first and second systolic peaks). This characteristic of the pulse pressure was described by the time-domain summation of two pairs of Gaussian-like waveforms (representing independent progressive and regressive components) with parameters related to amplitude, time lag, and duration for each component. A steepest descent optimization routine was used to fit the model parameters to experimental data of normotensive and subjects with hypertension. The optimized parameters were used to calculate two indices, RIx_1,2 (second-to-first systolic peak ratio) and RIx_1,3 (first diastolic-to-first systolic ratio). The observed responses between groups suggest that RIx_1,2 is related to an endothelial response to the ischemic process and could be used as a clinical tool to assess endothelial function in hypertension.     

A data-driven model to study utero-ovarian blood flow physiology during pregnancy

In this paper, we describe a mathematical model of the cardiovascular system in human pregnancy. An automated, closed-loop 1D–0D modelling framework was developed, and we demonstrate its efficacy in (1) reproducing measured multi-variate cardiovascular variables (pulse pressure, total peripheral resistance and cardiac output) and (2) providing automated estimates of variables that have not been measured (uterine arterial and venous blood flow, pulse wave velocity, pulsatility index). This is the first model capable of estimating volumetric blood flow to the uterus via the utero-ovarian communicating arteries. It is also the first model capable of capturing wave propagation phenomena in the utero-ovarian circulation, which are important for the accurate estimation of arterial stiffness in contemporary obstetric practice. The model will provide a basis for future studies aiming to elucidate the physiological mechanisms underlying the dynamic properties (changing shapes) of vascular flow waveforms that are observed with advancing gestation. This in turn will facilitate the development of methods for the earlier detection of pathologies that have an influence on vascular structure and behaviour.

     

On-line analysis of neuromuscular bioelectric potentials

A computer system is presented which provides for on-line data capture and analysis of evoked end-plate potentials and action potentials, and on-line data capture with off-line analysis of spontaneously occurring miniature end-plate potentials at the end-plate region of the neuromuscular junction. Sampling of evoked waveforms begins after an adjustable delay following the stimulus. Spontaneously occurring waveforms are captured by 'freezing' the contents of a circular buffer. The software provides MENU selectable support functions including storage and retrieval of data and calculated parameters, analog and digital display of waveforms, data calibration and gain modification, data editing, file management, and hardcopy output. Calculated parameters of the waveform are optionally placed in a data base file by the analysis programs. The data base may be used for editing, arithmetic operations, and subsetting of variables as well as statistical analysis and plotting of any selected variables.     

A mathematical model of maternal vascular growth and remodeling and changes in maternal hemodynamics in uncomplicated pregnancy

The maternal vasculature undergoes tremendous growth and remodeling (G&R) that enables a?>?15-fold increase in blood flow through the uterine vasculature from conception to term. Hemodynamic metrics (e.g., uterine artery pulsatility index, UA-PI) are useful for the prognosis of pregnancy complications; however, improved characterization of the maternal hemodynamics is necessary to improve prognosis. The goal of this paper is to develop a mathematical framework to characterize maternal vascular G&R and hemodynamics in uncomplicated human pregnancies. A validated 1D model of the human vascular tree from the literature was adapted and inlet blood flow waveforms at the ascending aorta at 4 week increments from 0 to 40 weeks of gestation were prescribed. Peripheral resistances of each terminal vessel were adjusted to achieve target flow rates and mean arterial pressure at each gestational age. Vessel growth was governed by wall shear stress (and axial lengthening in uterine vessels), and changes in vessel distensibility were related to vessel growth. Uterine artery velocity waveforms generated from this model closely resembled ultrasound results from the literature. The literature UA-PI values changed significantly across gestation, increasing in the first month of gestation, then dramatically decreasing from 4 to 20 weeks. Our results captured well the time-course of vessel geometry, material properties, and UA-PI. This 1D fluid-G&R model captured the salient hemodynamic features across a broad range of clinical reports and across gestation for uncomplicated human pregnancy. While results capture available data well, this study highlights significant gaps in available data required to better understand vascular remodeling in pregnancy.

     

Model for post-occlusive reactive hyperemia as measured noninvasively with pressure pulse waveform

Post-occlusive reactive hyperemia is a noninvasive maneuver to assess microvascular reactivity related to the bioavailability and/or bioactivity of endothelial-derived factors. The inability to respond to endogenous vasodilator substances is mostly described by a low peak flow after an event associated with a peak flow. The aim of this study is to propose a model to describe post-occlusive responses observed in the pressure waveforms after occlusion release. Model variables were investigated in search of those representatives of the endothelial response to the ischemic process. Radial pressure pulse waveforms were acquired in the anterior region of the wrist, superficial to the radial artery, using a piezoelectric transducer acquired by a 12 bits acquisition board model at a sampling rate of 1.0 kHz to increase the temporal resolution. The occlusion maneuver was performed using an arm-cuff placed over the brachial artery. A time series of pulse pressure (PP) values, calculated from successive values of beat-to-beat systolic and diastolic pressures, was found to be a useful variable representing blood pressure signal in the model. This data time series of the pulse pressure presents reduced initial values compared with the baseline measurement, and an increasing value until a steady state behavior was sustained after approximately 60 s. This behavior for the pulse pressure series was described by a hyperbolic tangent model with parameters K (rate of change of PP), PP₀ (first value of PP after cuff release), and ΔPP (change in PP). The model was applied to pulse pressure signals from normotensive and hypertensive subjects. The observed responses between groups suggest that PP₀ and ΔPP are related to an endothelial response to the ischemic process and could be used as a clinical tool to assess endothelial function in hypertension.     

Estimating the efficiency of cell capture and arrest in flow chambers: study of neutrophil binding via E-selectin and ICAM-1

A mathematical model was developed to quantify the efficiency of cell-substrate attachment in the parallel-plate flow chamber. The model decouples the physical features of the system that affect cell-substrate collision rates from the biological features that influence cellular adhesivity. Thus, experimental data on cell rolling and adhesion density are converted into "frequency" parameters that quantify the "efficiency" with which cells in the flow chamber progress from the free stream to rolling, and transition from rolling to firm arrest. The model was partially validated by comparing simulation results with experiments where neutrophils rolled and adhered onto substrates composed of cotransfected cells bearing E-selectin and intercellular adhesion molecule-1 (ICAM-1). Results suggest that: 1) Neutrophils contact the E-selectin substrate on average for 4-8.5s before tethering. This contact duration is insensitive to applied shear stress. 2) At 2 dyn/cm(2), approximately 28% of the collisions between the cells and substrate result in primary capture. Also, approximately 5-7% of collisions between neutrophils in the free stream and previously recruited neutrophils bound on the substrate result in secondary capture. These percentages were higher at lower shears. 3) An adherent cell may influence the flow streams in its vicinity up to a distance of 2.5 cell diameters away. 4) Our estimates of selectin on-rate in cellular systems compare favorably with data from reconstituted systems with immobilized soluble E-selectin. In magnitude, the observed on-rates occur in the order, L-selectin > P-selectin > E-selectin.     

Breakthrough performance of linear-DNA on ion-exchange membrane columns

Breakthrough performance of linear-DNA adsorption on ion-exchange membrane columns was theoretically and experimentally investigated using batch and fixed-bed systems. System dispersion curves showed the absence of flow non-idealities in the experimental arrangement. Breakthrough curves were not significantly affected by flow-rate or inlet solution concentration. In the theoretical analysis a model was integrated by the serial coupling of the membrane transport model and the system dispersion model. A transport model that considers finite kinetic rate and column dispersed flow was used in the study. A simplex optimization routine coupled to the solution of the partial differential model equations was employed to estimate the maximum adsorption capacity constant, the equilibrium desorption constant and the forward interaction rate-constant, which are the parameters of the membrane transport model. Through this approach a good prediction of the adsorption phenomena is obtained for inlet concentrations and flow rates greater than 0.2 mg/ml and 0.16 ml/min.     

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.     

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

     

Aortic velocity contours at abdominal branches in anesthetized dogs

Pulsed Doppler velocimetry was used to record a grid of velocity waveforms 0.5 cm downstream of the cephalic mesenteric and left renal branches of the aorta in anesthetized dogs. Aortic velocity contours at different phases of the pulse cycle were developed from the grid of waveforms. Transient flow recirculation occurred in the post-systolic phase of the pulse cycle at the aortic wall opposite the branch artery. There was no recirculation at the systolic peak but there was asymmetry of the velocity profile. In contrast, in the in vivo post-stenotic velocity field recirculation persists throughout most of the cycle. These results compare well with similar results from pulsatile model studies. Spectral width of the Doppler signal was not increased in any of the velocity waveforms immediately downstream of the branches, this suggests that flow is relatively undisturbed in these locations.     

A novel method for non-invasively detecting the severity and location of aortic aneurysms

The influence of an aortic aneurysm on blood flow waveforms is well established, but how to exploit this link for diagnostic purposes still remains challenging. This work uses a combination of experimental and computational modelling to study how aneurysms of various size affect the waveforms. Experimental studies are carried out on fusiform-type aneurysm models, and a comparison of results with those from a one-dimensional fluid–structure interaction model shows close agreement. Further mathematical analysis of these results allows the definition of several indicators that characterize the impact of an aneurysm on waveforms. These indicators are then further studied in a computational model of a systemic blood flow network. This demonstrates the methods’ ability to detect the location and severity of an aortic aneurysm through the analysis of flow waveforms in clinically accessible locations. Therefore, the proposed methodology shows a high potential for non-invasive aneurysm detectors/monitors.     

Computational assessment of model-based wave separation using a database of virtual subjects

The quantification of arterial wave reflection is an important area of interest in arterial pulse wave analysis. It can be achieved by wave separation analysis (WSA) if both the aortic pressure waveform and the aortic flow waveform are known. For better applicability, several mathematical models have been established to estimate aortic flow solely based on pressure waveforms. The aim of this study is to investigate and verify the model-based wave separation of the ARCSolver method on virtual pulse wave measurements.The study is based on an open access virtual database generated via simulations. Seven cardiac and arterial parameters were varied within physiological healthy ranges, leading to a total of 3325 virtual healthy subjects. For assessing the model-based ARCSolver method computationally, this method was used to perform WSA based on the aortic root pressure waveforms of the virtual patients. As a reference, the values of WSA using both the pressure and flow waveforms provided by the virtual database were taken.The investigated parameters showed a good overall agreement between the model-based method and the reference. Mean differences and standard deviations were −0.05 ± 0.02 AU for characteristic impedance, −3.93 ± 1.79 mmHg for forward pressure amplitude, 1.37 ± 1.56 mmHg for backward pressure amplitude and 12.42 ± 4.88% for reflection magnitude.The results indicate that the mathematical blood flow model of the ARCSolver method is a feasible surrogate for a measured flow waveform and provides a reasonable way to assess arterial wave reflection non-invasively in healthy subjects.     

Endothelial cell capture of heparin-binding growth factors under flow

Circulation is an important delivery method for both natural and synthetic molecules, but microenvironment interactions, regulated by endothelial cells and critical to the molecule's fate, are difficult to interpret using traditional approaches. In this work, we analyzed and predicted growth factor capture under flow using computer modeling and a three-dimensional experimental approach that includes pertinent circulation characteristics such as pulsatile flow, competing binding interactions, and limited bioavailability. An understanding of the controlling features of this process was desired. The experimental module consisted of a bioreactor with synthetic endothelial-lined hollow fibers under flow. The physical design of the system was incorporated into the model parameters. The heparin-binding growth factor fibroblast growth factor-2 (FGF-2) was used for both the experiments and simulations. Our computational model was composed of three parts: (1) media flow equations, (2) mass transport equations and (3) cell surface reaction equations. The model is based on the flow and reactions within a single hollow fiber and was scaled linearly by the total number of fibers for comparison with experimental results. Our model predicted, and experiments confirmed, that removal of heparan sulfate (HS) from the system would result in a dramatic loss of binding by heparin-binding proteins, but not by proteins that do not bind heparin. The model further predicted a significant loss of bound protein at flow rates only slightly higher than average capillary flow rates, corroborated experimentally, suggesting that the probability of capture in a single pass at high flow rates is extremely low. Several other key parameters were investigated with the coupling between receptors and proteoglycans shown to have a critical impact on successful capture. The combined system offers opportunities to examine circulation capture in a straightforward quantitative manner that should prove advantageous for biologicals or drug delivery investigations.     

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.