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.     

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.     

Validation of numerical flow simulations against in vitro phantom measurements in different type B aortic dissection scenarios

An aortic dissection (AD) is a serious condition defined by the splitting of the arterial wall, thus generating a secondary lumen [the false lumen (FL)]. Its management, treatment and follow-up are clinical challenges due to the progressive aortic dilatation and potentially severe complications during follow-up. It is well known that the direction and rate of dilatation of the artery wall depend on haemodynamic parameters such as the local velocity profiles, intra-luminal pressures and resultant wall stresses. These factors act on the FL and true lumen, triggering remodelling and clinical worsening. In this study, we aimed to validate a computational fluid dynamic (CFD) tool for the haemodynamic characterisation of chronic (type B) ADs. We validated the numerical results, for several dissection geometries, with experimental data obtained from a previous in vitro study performed on idealised dissected physical models. We found a good correlation between CFD simulations and experimental measurements as long as the tear size was large enough so that the effect of the wall compliance was negligible.     

Hemodynamics in the carotid artery bifurcation: a comparison between numerical simulations and in vitro MRI measurements

The presence of atherosclerotic plaques has been shown to be closely related to the vessel geometry. Studies on postmortem human arteries and on the experimental animal show positive correlation between the presence of plaque thickness and low shear stress, departure of unidirectional flow and regions of flow separation and recirculation. Numerical simulations of arterial blood flow and direct blood flow velocity measurements by magnetic resonance imaging (MRI) are two approaches for the assessment of arterial blood flow patterns. In order to verify that both approaches give equivalent results magnetic resonance velocity data measured in a compliant anatomical carotid bifurcation model were compared to the results of numerical simulations performed for a corresponding computational vessel model. Cross sectional axial velocity profiles were calculated and measured for the midsinus and endsinus internal carotid artery. At both locations a skewed velocity profile with slow velocities at the outer vessel wall, medium velocities at the side walls and high velocities at the flow divider (inner) wall were observed. Qualitative comparison of the axial velocity patterns revealed no significant differences between simulations and in vitro measurements. Even quantitative differences such as for axial peak flow velocities were less than 10%. Secondary flow patterns revealed some minor differences concerning the form of the vortices but maximum circumferential velocities were in the same range for both methods.     

Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations

In this paper we present a macroscopic model of the excitation process in the myocardium. The composite and anisotropic structure of the cardiac tissue is represented by a bidomain, i.e. a set of two coupled anisotropic media. The model is characterized by a non linear system of two partial differential equations of parabolic and elliptic type. A singular perturbation analysis is carried out to investigate the cardiac potential field and the structure of the moving excitation wavefront. As a consequence the cardiac current sources are approximated by an oblique dipole layer structure and the motion of the wavefront is described by eikonal equations. Finally numerical simulations are carried out in order to analyze some complex phenomena related to the spreading of the wavefront, like the front-front or front-wall collision. The results yielded by the excitation model and the eikonal equations are compared.     

Numerical simulation of wave propagation in a realistic model of the human external ear

In this study, a numerical investigation is performed to evaluate the effects of high-pressure sinusoidal and blast wave's propagation around and inside of a human external ear. A series of computed tomography images are used to reconstruct a realistic three-dimensional (3D) model of a human ear canal and the auricle. The airflow field is then computed by solving the governing differential equations in the time domain using a computational fluid dynamics software. An unsteady algorithm is used to obtain the high-pressure wave propagation throughout the ear canal which is validated against the available analytical and numerical data in literature. The effects of frequency, wave shape, and the auricle on pressure distribution are then evaluated and discussed. The results clearly indicate that the frequency plays a key role on pressure distribution within the ear canal. At 4 kHz frequency, the pressure magnitude is much more amplified within the ear canal than the frequencies of 2 and 6 kHz, for the incident wave angle of 90° investigated in this study, attributable to the ‘4-kHz notch’ in patients with noise-induced hearing loss. According to the results, the pressure distribution patterns at the ear canal are very similar for both sinusoidal pressure waveform with the frequency of 2 kHz and blast wave. The ratio of the peak pressure value at the eardrum to that at the canal entrance increases from about 8% to 30% as the peak pressure value of the blast wave increases from 5 to 100 kPa for the incident wave angle of 90° investigated in this study. Furthermore, incorporation of the auricle to the ear canal model is associated with centerline pressure magnitudes of about 50% and 7% more than those of the ear canal model without the auricle throughout the ear canal for sinusoidal and blast waves, respectively, without any significant effect on pressure distribution pattern along the ear canal for the incident wave angle of 90° investigated in this study.     

Wave propagation in a model of the arterial circulation

The propagation of the arterial pulse wave in the large systemic arteries has been calculated using a linearised method of characteristics analysis to follow the waves generated by the heart. The model includes anatomical and physiological data for the 55 largest arteries adjusted so that the bifurcating tree of arteries is well matched for forward travelling waves. The peripheral arteries in the model are terminated by resistance elements which are adjusted to produce a physiologically reasonable distribution of mean blood flow. In the model, the pressure and velocity wave generated by the contraction of the left ventricle propagates to the periphery where it is reflected. These reflected waves are re-reflected by each of the bifurcations that they encounter and a very complex pattern of waves is generated. The results of the calculations exhibit many of the features of the systemic arteries, including the increase of the pulse pressure with distance away from the heart as well as the initial decrease and then the large increase in the magnitude of back flow during late systole going from the ascending aorta to the abdominal aorta to the arteries of the leg. The model is then used to study the effects of the reflection or absorption of waves by the heart and the mechanisms leading to the incisura are investigated. Calculations are carried out with the total occlusion of different arterial segments in order to model experiments in which the effects of the occlusion of different arteries on pressure and flow in the ascending aorta were measured. Finally, the effects of changes in peripheral resistance on pressure and velocity waveforms are also studied. We conclude from these calculations that the complex pattern of wave propagation in the large arteries may be the most important determinant of arterial haemodynamics.     

Theoretical analysis of the magnetocardiographic pattern for reentry wave propagation in a three-dimensional human heart model

We present a computational study of reentry wave propagation using electrophysiological models of human cardiac cells and the associated magnetic field map of a human heart. We examined the details of magnetic field variation and related physiological parameters for reentry waves in two-dimensional (2-D) human atrial tissue and a three-dimensional (3-D) human ventricle model. A 3-D mesh system representing the human ventricle was reconstructed from the surface geometry of a human heart. We used existing human cardiac cell models to simulate action potential (AP) propagation in atrial tissue and 3-D ventricular geometry, and a finite element method and the Galerkin approximation to discretize the 3-D domain spatially. The reentry wave was generated using an S1-S2 protocol. The calculations of the magnetic field pattern assumed a horizontally layered conductor for reentry wave propagation in the 3-D ventricle. We also compared the AP and magnetocardiograph (MCG) magnitudes during reentry wave propagation to those during normal wave propagation. The temporal changes in the reentry wave motion and magnetic field map patterns were also analyzed using two well-known MCG parameters: the current dipole direction and strength. The current vector in a reentry wave forms a rotating spiral. We delineated the magnetic field using the changes in the vector angle during a reentry wave, demonstrating that the MCG pattern can be helpful for theoretical analysis of reentry waves.     

Radon transfer from thermal water to human organs in radon therapy: exhalation measurements and model simulations

Radiation and Environmental Biophysics - The transfer of radon from thermal water via the skin to different human organs in radon therapy can experimentally be determined by measuring the radon...     

A general model for the propagation of uncertainty in measurements into heat transfer simulations and its application to cryosurgery

This report presents a technique for estimating the propagation of uncertainty in measurements into mathematical simulations of heat transfer. The motivation for this report is to show the dramatic uncertainty associated with estimating the value of the so-called "lethal temperature," even in a case where a perfect correlation appears to exist between histo-pathologic observations and a corresponding heat transfer simulation. Although the example presented in this report relates to cryosurgery, the technique proposed in this report is rather general and can be applied to any heat transfer problem. The uncertainty analysis presented in this report can be considered as an extension of the well-known concept of the rule of the square root of the sum of the square errors. A comparison of the new technique with the worst case scenario concept is also presented. In conclusion, it is recommended that the proposed technique be routinely applied when presenting simulated results, whether as a part of a theoretical study, or in comparison with experimental data.     

Formulation and numerical simulations of a continuum model of avascular tumor growth

In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G₀ phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.     

An unconditionally stable numerical method for the Luo-Rudy 1 model used in simulations of defibrillation

Numerical simulations of defibrillation using the Bidomain model coupled to a model of membrane kinetics represent a serious numerical challenge. This is because very high voltages close to defibrillation electrodes demand that extreme time step restrictions be placed on standard numerical schemes, e.g. the forward Euler scheme. A common solution to this problem is to modify the cell model by simple if-tests applied to several equations and rate functions. These changes are motivated by numerical problems rather than physiology, and should therefore be avoided whenever possible. The purpose of this paper is to present a numerical scheme that handles the original model without modifications and which is unconditionally stable for the Luo-Rudy phase 1 model. This also shows that the cell model is mathematically well-behaved, even in the presence of very high voltages. Our theoretical results are illustrated by numerical computations.     

Intersegmental coordination in the lamprey: simulations using a network model without segmental boundaries

Swimming in vertebrates such as eel and lamprey involves the coordination of alternating left and right activity in each segment. Forward swimming is achieved by a lag between the onset of activity in consecutive segments rostrocaudally along the spinal cord. The intersegmental phase lag is approximately 1% of the cycle duration per segment and is independent of the swimming frequency. Since the lamprey has approximately 100 spinal segments, at any given time one wave of activity is propagated along the body. Most previous simulations of intersegmental coordination in the lamprey have treated the cord as a chain of coupled oscillators or well-defined segments. Here a network model without segmental boundaries is described which can produce coordinated activity with a phase lag. This ‘continuous’ pattern-generating network is composed of a column of 420 excitatory interneurons (E1 to E420) and 300 inhibitory interneurons (C1 to C300) on each half of the simulated spinal cord. The interneurons are distributed evenly along the simulated spinal cord, and their connectivity is chosen to reflect the behavior of the intact animal and what is known about the length and strength of the synaptic connections. For example, E100 connects to all interneurons between E51 and E149, but at varying synaptic strengths, while E101 connects to all interneurons between E52 and E150. This unsegmented E-C network generates a motor pattern that is sampled by output elements similar to motoneurons (M cells), which are arranged along the cell column so that they receive input from seven E and five C interneurons. The M cells thus represent the summed excitatory and inhibitory input at different points along the simulated spinal cord and can be regarded as representing the ventral root output to the myotomes along the spinal cord. E and C interneurons have five simulated compartments and Hodgkin-Huxley based dynamics. The simulated network produces rhythmic output over a wide range of frequencies (1–11 Hz) with a phase lag constant over most of the length, with the exception of the ‘cut’ ends due to reduced synaptic input. As the inhibitory C interneurons in the simulation have more extensive caudal than rostral projections, the output of the simulation has positive phase lags, as occurs in forward swimming. However, unlike the biological network, phase lags in the simulation increase significantly with burst frequency, from 0.5% to 2.3% over the range of frequencies of the simulation. Local rostral or caudal increases in excitatory drive in the simulated network are sufficient to produce motor patterns with increased or decreased phase lags, respectively. Received: 15 December 1995 / Accepted in revised form: 17 September 1996     

Pulse wave velocity as a diagnostic index: the pitfalls of tethering versus stiffening of the arterial wall

Pulse wave velocity (PWV) is often used as a clinical index of aging, vascular disease, or age related hypertension. This practice is based on the assumption that a higher wave speed indicates vascular stiffening. This assumption is well grounded in the physics of pulsatile flow of an incompressible fluid where it is fully established that a pulse wave travels faster in a tube of stiffer wall, the wave speed becoming infinite in the mathematical limit of a rigid wall. However, in this paper we point out that the physical principal of higher pulse wave velocity in a stiffer tube is strictly valid only when the wall is free from outside constraints, which in the physiological setting is present in the form of tethering of the vessel wall. The use of PWV as an index of arterial stiffening may thus lose its validity if tethering is involved. A solution of the problem of vessel wall mechanics as they arise from the physiological pulsatile flow problem is presented for the purpose of resolving this issue. The vessel wall is considered to have finite thickness with or without tethering and with a range of mechanical properties ranging from viscoelastic to stiff. The results show that, indeed, while the wave speed becomes infinite in the mathematical limit of a rigid free wall, the opposite actually happens if the vessel wall is tethered. Here the wave speed actually diminishes as the degree of tethering increases. This dichotomy in the effects of tethering versus stiffening of the arterial wall may clearly lead to error in the interpretation of PWV as an index of vessel wall stiffness. In particular, a normal value of PWV may lead to the conclusion that vessel wall stiffening is absent while this value may in fact have been lowered by tethering. In other words, the diagnostic test may lead to a false negative diagnosis. Our results indicate that the reason for which PWV is lower in a tethered wall compared with that in a free wall of the same stiffness is that the radial movements of the wall are greatly reduced by tethering. More precisely, the results show that PWV depends strongly on the ratio of radial to axial displacements and that this ratio is much lower in a tethered wall than it is in a free wall of the same stiffness.     

LLC-PK1 epithelia as a model for in vitro assessment of proximal tubular nephrotoxicity

Summary LLC-PK₁ cells, an established epithelial cell line derived from pig kidney, were used as a model system for assessment of nephrotoxic side effects of three cephalosporin antibiotics: cephaloridine, ceftazidime, and cefotaxime. Toxic effects of these xenobiotics were monitored on confluent monolayers by light and electron microscopy and by the release of cellular marker enzyme activities into the culture medium. In addition, LLC-PK₁ cells were grown on microporous supports, and cephalosporin-induced alteration of epithelial functional integrity was monitored by a novel electrophysiologic approach. For this purpose, an Ussing chamberlike experimental setup was used. The dose-dependent effects on transepithelial ionic permselectivity were monitored under conditions in which defined fractions of the apical culture medium NaCl contents were replaced iso-osmotically by mannitol. This method of determining the functional intactness of the epithelial barrier by measuring dilution potentials was found to be far more sensitive than monitoring cell injury by means of morphology or measurement of enzyme release. As expected from animal experimental data, a dose-dependent disruption of monolayer integrity was detected with all three methodologies applied. Cephaloridine was found the most toxic compound followed by ceftazidime, where a 3-fold, and cefotaxime, where a 10-fold dose of that of cephaloridine was needed to produce cell injury. Measurement of transepithelial dilution potentials was more sensitive as compared to the release of the apical plasma membrane marker enzyme activities alkaline phosphatase andγ-glutamyltranspeptidase, the cytosolic lactate dehydrogenase, or the mitochondrial glutamate dehydrogenase. The data were compared to the effects of the aminoglycoside antibiotic gentamicin, which at least with respect to its effects on LLC-PK₁ morphology and enzyme release, but not transepithelial electrical properties, was already investigated.     

Mathematical model of activation wave propagation in an isolated cardiomyocyte

In order to describe spontaneous wave-like contractions of a single isolated cardiomyocyte a mathematical model is proposed, which relates this phenomenon to propagation of calcium ion concentration wave along the cell. Free diffusion of Ca2+ ions as well as their reversible binding to regulatory proteins in contractile apparatus, Ca2+ accumulation in sarcoplasmic reticulum, and Ca-induced Ca2+ release are included in the governing equations. The model agrees with some observations. It predicts also some effects which may by a subject of future experimental research.     

Serial propagation of human endothelial cells in vitro

Human umbilical vein (HUV) endothelial cells were grown for 15 to 21 passages at a split ratio of 1:5 (at least 27 population doublings) on a human fibronectin (HFN) matrix in Medium 199 supplemented with fetal bovine serum (FBS) and endothelial-cell growth factor (ECGF). This system also permitted the growth of HUV endothelial cells at cell densities as low as 1.25 cells/cm2. In addition to delaying the premature senescence of HUV endothelial cells, ECGF also reduced the serum requirement for low-density HUV endothelial-cell growth; 2.5% serum and ECGF yields half-maximum growth as compared to high serum controls. Significant HUV endothelial-cell growth was also observed in medium supplemented with either ovine hypophysectomized (HYPOX) serum, plasma-derived serum (PDS), or HYPOX-PDS in the presence of ECGF, suggesting that neither the pituitary nor the platelet contributes to HUV endothelial-cell growth.