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.     

Theoretical analysis of occlusion techniques for measuring pulmonary capillary pressure.

We have developed a model including three serial compliant compartments (arterial, capillary, and venous) separated by two resistances (arterial and venous) for interpreting in vivo single pulmonary arterial or venous occlusion pressure profiles and double occlusion. We formalized and solved the corresponding system of equations. We showed that in this model 1) pulmonary capillary pressure (Pc) profile after arterial or venous occlusion has an S shape, 2) the estimation of Pc by zero time extrapolation of the slow component of the arterial occlusion profile (Pcao) always overestimates Pc, 3) symmetrically such an estimation on the venous occlusion profile (Pcvo) always underestimates Pc, 4) double occlusion pressure (Pcdo) differs from Pc. We evaluated the impact of varying parameter values in the model with parameter sets drawn either from the literature or from arbitrary arterial and venous pressures, being respectively 20 and 5 mmHg. Resulting Pcao-Pc differences ranged from 0.4 to 5.4 mmHg and resulting Pcvo-Pc differences ranged from -0.3 to -5.0 mmHg. Pcdo-Pc was positive or negative, its absolute value in general being negligible (< 1.1 mmHg).     

Modeling baroreflex regulation of heart rate during orthostatic stress

During orthostatic stress, arterial and cardiopulmonary baroreflexes play a key role in maintaining arterial pressure by regulating heart rate. This study presents a mathematical model that can predict the dynamics of heart rate regulation in response to postural change from sitting to standing. The model uses blood pressure measured in the finger as an input to model heart rate dynamics in response to changes in baroreceptor nerve firing rate, sympathetic and parasympathetic responses, vestibulo-sympathetic reflex, and concentrations of norepinephrine and acetylcholine. We formulate an inverse least squares problem for parameter estimation and successfully demonstrate that our mathematical model can accurately predict heart rate dynamics observed in data obtained from healthy young, healthy elderly, and hypertensive elderly subjects. One of our key findings indicates that, to successfully validate our model against clinical data, it is necessary to include the vestibulo-sympathetic reflex. Furthermore, our model reveals that the transfer between the nerve firing and blood pressure is nonlinear and follows a hysteresis curve. In healthy young people, the hysteresis loop is wide, whereas, in healthy and hypertensive elderly people, the hysteresis loop shifts to higher blood pressure values, and its area is diminished. Finally, for hypertensive elderly people, the hysteresis loop is generally not closed, indicating that, during postural change from sitting to standing, baroreflex modulation does not return to steady state during the first minute of standing.     

Pressure and flow in the systemic arterial system

The dynamic characteristics of the proximal arterial system are studied by solving the nonlinear momentum and mass conservation equations for pressure and flow. The equations are solved for a model systemic arterial system that includes the aorta, common iliacs, and the internal and external iliac arteries. The model includes geometric and elastic taper of the aorta, nonlinearly elastic arteries, side flows, and a complex distal impedance. The model pressure wave shape, inlet and outlet impedance, wave travel, and apparent wave velocity compare favorably with the values measured on humans. Calculations indicate that: (i) reflections are the major factor determining the shape and distal amplification of the pressure wave in the arterial tree; (ii) although important in attenuating the proximal transmission of reflecting waves, geometric taper is not the major cause of the distal pressure wave amplification; (iii) the dicrotic wave is a result of peripheral reflection and is not due to the sudden change in flow at the end of systole; (iv) the elastic taper and nonlinearity of the wall elasticity are of minor significance in determining the flow and pressure profiles; and (v) in spite of numerous nonlinearities, the system behaves in a somewhat linear fashion for the lower frequency components.     

An agent-based model of group decision making in baboons

We present an agent-based model of the key activities of a troop of chacma baboons (Papio hamadryas ursinus) based on the data collected at De Hoop Nature Reserve in South Africa. We analyse the predictions of the model in terms of how well it is able to duplicate the observed activity patterns of the animals and the relationship between the parameters that control the agent's decision procedure and the model's predictions. At the current stage of model development, we are able to show that across a wide range of decision parameter values, the baboons are able to achieve their energetic and social time requirements. The simulation results also show that decisions concerning movement (group action selection) have the greatest influence on the outcomes. Those cases where the model's predictions fail to agree with the observed activity patterns have highlighted key elements that were missing from the field data, and that would need to be collected in subsequent fieldwork. Based on our experience, we believe group decision making is a fertile field for future research, and agent-based modelling offers considerable scope for understanding group action selection.     

Individual variability of cardiovascular reactivity during protective cardiac reflex in humans

Fail of fast recovering to normal state after aversive event is the key feature of individual intolerance to emotional stress. Within the framework of this idea, dynamics of the arterial blood pressure reactivity was studied studied using defensive reaction of the cardiac defense response elicited by strong aversive acoustic stimuli. Dynamical patterns of cardiovascular responses observed in this reaction makes made it possible to perform affective chronometry of defensive arousal. Healthy male volunteers with normal arterial blood pressure and patients with firstly diagnosed and untreated essential hypertension were studied. The beat-by-beat dynamics of blood pressure values were registered by means of finger-cuff Finapres technology. It was shown that among healthy individuals with normal resting blood pressure values hyperreactive persons with delayed late systolic and diastolic blood pressure increases ocurred. It was suggested that similiarity of delayed pressor effects ofhypereactive healthy individuals and those of hypertension patients incate a high risk of essential hypertension in highly reactive individuals.     

Three-section transmission-line arterial model for noninvasive assessment of vascular remodeling in primary hypertension

High central arterial blood pressure can be sustained by the capacity of living arteries to respond to hemodynamic stimuli by changing their structural and/or functional characteristics. These adaptations are considered to occur in a time-dependency, in which different patterns of vascular geometry are identified at all stages. This paper proposes a three-section transmission-line model of the brachial-radial arterial segment and a rational procedure to analyze its transfer function that can be used to interpret the longitudinal remodeling process of medium-sized arteries. The three sections of the model correspond to different arterial segments of the forearm. The model processed pressure signals collected noninvasively from normotensive and hypertensive volunteers at brachial and radial arteries. Aiming to explain possible hypertrophic inward remodeling, geometrical model parameters obtained from normotensive individuals were modified in order to generate high-pressure pulses observed in the hypertensive subjects. The resulting transfer functions for the hypertrophy adaptation exhibit properties related to the pathophysiology of the remodeling process, mainly the reduced amplification of the higher harmonics of the pulse waveform. The results suggest the model can be used to assess noninvasively the hypertension-induced adaptations related to geometrical characteristics of the medium-size arteries.     

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.     

Cerebral hemodynamics during arterial and CO(2) pressure changes: in vivo prediction by a mathematical model

The aim of this work was to analyze changes in cerebral hemodynamics and intracranial pressure (ICP) evoked by mean systemic arterial pressure (SAP) and arterial CO(2) pressure (Pa(CO(2))) challenges in patients with acute brain damage. The study was performed by means of a new simple mathematical model of intracranial hemodynamics, particularly aimed at routine clinical investigation. The model was validated by comparing its results with data from transcranial Doppler velocity in the middle cerebral artery (V(MCA)) and ICP measured in 44 tracings on 13 different patients during mean SAP and Pa(CO(2)) challenges. The validation consisted of individual identification of 6 parameters in all 44 tracings by means of a best fitting algorithm. The parameters chosen for the identification summarize the main aspects of intracranial dynamics, i.e., cerebrospinal fluid circulation, intracranial elastance, and cerebrovascular control. The results suggest that the model is able to reproduce the measured time patterns of V(MCA) and ICP in all 44 tracings by using values for the parameters that lie within the ranges reported in the pathophysiological literature. The meaning of parameter estimates is discussed, and comments on the main virtues and limitations of the present approach are offered.     

Abnormal arterial flows by a distributed model of the fetal circulation

Modeling the propagation of blood pressure and flow along the fetoplacental arterial tree may improve interpretation of abnormal flow velocity waveforms in fetuses. The current models, however, either do not include a wide range of gestational ages or do not account for variation in anatomical, vascular, or rheological parameters. We developed a mathematical model of the pulsating fetoumbilical arterial circulation using Womersley's oscillatory flow theory and viscoelastic arterial wall properties. Arterial flow waves are calculated at different arterial locations from which the pulsatility index (PI) can be determined. We varied blood viscosity, placental and brain resistances, placental compliance, heart rate, stiffness of the arterial wall, and length of the umbilical arteries. The PI increases in the umbilical artery and decreases in the cerebral arteries, as a result of increasing placental resistance or decreasing brain resistance. Both changes in resistance decrease the flow through the placenta. An increased arterial stiffness increases the PIs in the entire fetoplacental circulation. Blood viscosity and peripheral bed compliance have limited influence on the flow profiles. Bradycardia and tachycardia increase and decrease the PI in all arteries, respectively. Umbilical arterial length has limited influence on the PI but affects the mean arterial pressure at the placental cord insertion. The model may improve the interpretation of arterial flow pulsations and thus may advance both the understanding of pathophysiological processes and clinical management.     

Towards early diagnosis of atherosclerosis: the finite volume method for fluid-structure interaction

Blood flow through arteries represents a very complex, fluid-structure interaction (FSI) problem. Strong coupling between the blood and artery is due to the relatively low stiffness of the artery compared to that of blood. Hence, the pressure exerted by the flowing blood on the artery wall can result in considerable deformations of the artery, and vice-versa, arterial deformations can in turn affect the blood flow. In the present work, the finite volume method is employed to solve the problem where compressible fluid, representing blood, flows in healthy arteries as well as in unhealthy, i.e., partly stiffened arteries. The stiffening of the arterial wall is assumed to be the first key stage in the development of atherosclerosis. The comparison between various deformation profiles of healthy and unhealthy arteries demonstrates significant and measurable differences, in particular in the radial direction. This is hoped to help toward establishing procedures for early diagnosis of the disease.     

Universally sloppy parameter sensitivities in systems biology models

Quantitative computational models play an increasingly important role in modern biology. Such models typically involve many free parameters, and assigning their values is often a substantial obstacle to model development. Directly measuring in vivo biochemical parameters is difficult, and collectively fitting them to other experimental data often yields large parameter uncertainties. Nevertheless, in earlier work we showed in a growth-factor-signaling model that collective fitting could yield well-constrained predictions, even when it left individual parameters very poorly constrained. We also showed that the model had a “sloppy” spectrum of parameter sensitivities, with eigenvalues roughly evenly distributed over many decades. Here we use a collection of models from the literature to test whether such sloppy spectra are common in systems biology. Strikingly, we find that every model we examine has a sloppy spectrum of sensitivities. We also test several consequences of this sloppiness for building predictive models. In particular, sloppiness suggests that collective fits to even large amounts of ideal time-series data will often leave many parameters poorly constrained. Tests over our model collection are consistent with this suggestion. This difficulty with collective fits may seem to argue for direct parameter measurements, but sloppiness also implies that such measurements must be formidably precise and complete to usefully constrain many model predictions. We confirm this implication in our growth-factor-signaling model. Our results suggest that sloppy sensitivity spectra are universal in systems biology models. The prevalence of sloppiness highlights the power of collective fits and suggests that modelers should focus on predictions rather than on parameters.     

Tuning of pulmonary arterial circulation evidenced by MR phase mapping in healthy volunteers.

Magnetic resonance (MR) phase mapping was used to noninvasively assess both blood flow and cross-sectional area (CSA) in the main pulmonary artery (MPA) of 12 healthy volunteers. Flow and CSA patterns exhibited two positive peaks: high systolic and small diastolic. This finding can be explained using a simple "distributed" theoretical model that takes into account the role of a reflected pressure wave from pulmonary vascular impedance in generating a diastolic flow. The mean reflection coefficient of pressure wave, MPA input impedance, and pulmonary vascular impedance were assessed. We verified, in this series, that pressure wave velocity appears to be age-dependent. MR phase mapping has been used to observe the tuning (resonance) of the right cardiovascular system at rest under physiological conditions. MR phase mapping could be used to assess pathological modifications of the tuning that occurs in cases of pulmonary arterial hypertension.     

Localization of the sites of pulmonary vasomotion by use of arterial and venous occlusion

In this study, we present a new approach for using the pressure vs. time data obtained after various vascular occlusion maneuvers in pump-perfused lungs to gain insight into the longitudinal distribution of vascular resistance with respect to vascular compliance. Occlusion data were obtained from isolated dog lung lobes under normal control conditions, during hypoxia, and during histamine or serotonin infusion. The data used in the analysis include the slope of the arterial pressure curve and the zero time intercept of the extrapolated venous pressure curve after venous occlusion, the equilibrium pressure after simultaneous occlusion of both the arterial inflow and venous outflow, and the area bounded by equilibrium pressure and the arterial pressure curve after arterial occlusion. We analyzed these data by use of a compartmental model in which the vascular bed is represented by three parallel compliances separated by two series resistances, and each of the three compliances and the two resistances can be identified. To interpret the model parameters, we view the large arteries and veins as mainly compliance vessels and the small arteries and veins as mainly resistance vessels. The capillary bed is viewed as having a high compliance, and any capillary resistance is included in the two series resistances. With this view in mind, the results are consistent with the major response to serotonin infusion being constriction of large and small arteries (a decrease in arterial compliance and an increase in arterial resistance), the major response to histamine infusion being constriction of small and large veins (an increase in venous resistance and a decrease in venous compliance), and the major response to hypoxia being constriction of the small arteries (an increase in arterial resistance). The results suggest that this approach may have utility for evaluation of the sites of action of pulmonary vasomotor stimuli.     

New interpretation of arterial stiffening due to cigarette smoking using a structurally motivated constitutive model

Cigarette smoking is the leading self-inflicted risk factor for cardiovascular diseases; it causes arterial stiffening with serious sequelea including atherosclerosis and abdominal aortic aneurysms. This work presents a new interpretation of arterial stiffening caused by smoking based on data published for rat pulmonary arteries. A structurally motivated "four fiber family" constitutive relation was used to fit the available biaxial data and associated best-fit values of material parameters were estimated using multivariate nonlinear regression. Results suggested that arterial stiffening caused by smoking was reflected by consistent increase in an elastin-associated parameter and moreover by marked increase in the collagen-associated parameters. That is, we suggest that arterial stiffening due to cigarette smoking appears to be isotropic, which may allow simpler phenomenological models to capture these effects using a single stiffening parameter similar to the approach in isotropic continuum damage mechanics. There is a pressing need, however, for more detailed histological information coupled with more complete biaxial mechanical data for a broader range of systemic arteries.     

A physiological model for predicting carboxyhemoglobin formation from exposure to carbon monoxide in rats.

A time-dependent simulation model, based on the Coburn-Forster-Kane equation, was written in Advanced Continuous Simulation Language to predict carboxyhemoglobin (HbCO) formation and dissociation in F-344 rats during and after exposure to 500 parts/million CO for 1 h. Blood-gas analysis and CO-oximetry were performed on samples collected during exposure and off-gassing of CO. Volume displacement plethysmography was used to measure minute ventilation (VE) during exposure. CO diffusing capacity in the lung (DLCO) was also measured. Other model parameters measured in the animals included blood pH, total blood volume, and Hb concentration. Comparisons between model predictions using values for VE, DLCO, and the Haldane coefficient cited in the literature and predictions using measured VE, DLCO, and calculated Haldane coefficient for individual animals were made. General model predictions using values for model parameters derived from the literature agreed with published HbCO values by a factor of 0.987 but failed to simulate experimental data. On average, the general model overpredicted measured HbCO level by nearly 9%. A specific model using the means of measured variables predicted HbCO concentration within a factor of 0.993. When experimentally observed parameter fluctuations were included, the specific model predictions reflected experimental effects on HbCO formation.     

Impedance analysis of a tight epithelium using a distributed resistance model.

This paper develops techniques for equivalent circuit analysis of tight epithelia by alternating-current impedance measurements, and tests these techniques on rabbit urinary bladder. Our approach consists of measuring transepithelial impedance, also measuring the DC voltage-divider ratio with a microelectrode, and extracting values of circuit parameters by computer fit of the data to an equivalent circuit model. We show that the commonly used equivalent circuit models of epithelia give significant misfits to the impedance data, because these models (so-called "lumped models") improperly represent the distributed resistors associated with long and narrow spaces such as lateral intercellular spaces (LIS). We develop a new "distributed model" of an epithelium to take account of these structures and thereby obtain much better fits to the data. The extracted parameters include the resistance and capacitance of the apical and basolateral cell membranes, the series resistance, and the ratio of the cross-sectional area to the length of the LIS. The capacitance values yield estimates of real area of the apical and basolateral membranes. Thus, impedance analysis can yield morphological information (configuration of the LIS, and real membrane areas) about a living tissue, independently of electron microscopy. The effects of transport-modifying agents such as amiloride and nystatin can be related to their effects on particular circuit elements by extracting parameter values from impedance runs before and during application of the agent. Calculated parameter values have been validated by independent electrophysiological and morphological measurements.     

Influence of the distensibility of large arteries on the longitudinal impedance: application for the development of non-invasive techniques to the diagnosis of arterial diseases

Background

This study shows that the arterial longitudinal impedance constitutes a hemodynamic parameter of interest for performance characterization of large arteries in normal condition as well as in pathological situations. For this purpose, we solved the Navier?CStokes equations for an incompressible flow using the finite element analysis method and the Arbitrary Lagrangian Eulerian (ALE) formulation. The mathematical model assumes a two-dimensional flow and takes into account the nonlinear terms in the equations of fluid motion that express the convective acceleration, as well as the nonlinear deformation of the arterial wall. Several numerical simulations of the blood flow in large vessels have been performed to study the propagation along an arterial vessel of a pressure gradient pulse and a rate flow pulse. These simulations include various deformations of the wall artery leading to parietal displacements ranging from 0 (rigid wall) to 15% (very elastic wall) in order to consider physiological and pathological cases.

Results

The results show significant changes of the rate flow and the pressure gradient wave as a function of aosc, the relative variation in the radius of the artery over a cardiac cycle. These changes are notable beyond a critical value of aosc equal to 0.05. This critical value is also found in the evolution of the longitudinal impedance. So, above a variation of radius of 5%, the convective acceleration, created by the fluid-wall interactions, have an influence on the flow detectable on the longitudinal impedance.

Conclusions

The interpretation of the evolution of the longitudinal impedance shows that it could be a mean to test the performance of large arteries and can contribute to the diagnosis of parietal lesions of large arteries. For a blood vessel with a wall displacement higher than 5% similar to those of large arteries like the aorta, the longitudinal impedance is substantially greater than that obtained in the absence of wall displacement. This study also explains the effects of convective acceleration, on the shape of the decline of the pressure gradient wave and shows that they should not be neglected when the variation in radius is greater than 5%.     

Teleonomical optimization of a fractal model of the pulmonary arterial bed

Modeling the pulmonary arterial tree (PAT) is considered here as an optimal synthesis problem. Firstly, a class of candidate models is specified: the three-dimensional symmetric dichotomous fractal trees of elastic tubes described by Womersley's equations. Secondly, the parameters are shown to be constrained by interactions of PAT with the rest of the body; these constraints are used to limit the volume of the parametric space to which attention will be directed in the synthesis step. Thirdly, a teleonomical hypothesis is proposed: a naturally selected PAT must have a minimal input impedance under conditions keeping total arterial volume and distensibility as small as possible. This hypothesis is translated in mathematical terms and the resulting cost-function minimized in the limited parametric volume. The optimal model has parameter values and an impedance spectrum corresponding satisfactorily with real data. Moreover this model gives a clear picture of the internal hemodynamic behavior of PAT as an impedance matching device.     

Comparison of the hemodynamics in 6mm and 4-7 mm hemodialysis grafts by means of CFD

The aim of our study is to investigate with computational fluid dynamics (CFD) whether different arterial anastomotic geometries result in a different hemodynamics at the arterial (AA) and venous anastomosis (VA) of hemodialysis vascular access grafts. We have studied a 6mm graft (CD) and a 4-7 mm graft (TG). A validated three-dimensional CFD model is developed to simulate flow in the two graft types. Only the arterial anastomosis (AA) geometry differs. The boundary conditions applied are a periodic velocity signal at the arterial inlet and a periodic pressure wave at the venous outlet. Flow rate is set to 1,000 ml/min. The time dependent Navier-Stokes equations are solved. Wall shear stress (WSS), wall shear stress gradient (WSSG) and pressure gradient (PG) are calculated. Anastomotic flow is asymmetric although the anastomosis geometry is symmetric. The hemodynamic parameters, WSS, WSSG and PG, values at the suture line of the arterial anastomosis of the TG are at least twice as much as in the CD. Comparing the parameters at the two AA indicate that little flow rate increase introduces the risk of hemolysis in the TG whereas the CD is completely free of hemolysis. The hemodynamic parameter values at the venous anastomosis of the CD are 24 till 35% higher compared to the values of the TG. WSS values (> 3 Pa) in the VA are in the critical range for stenosis development in both graft geometries. The zones where the parameters reach extreme values correspond to the locations where intimal hyperplasia formation is reported in literature. In all anastomoses, the hemodynamic parameter levels are in the range where leucocytes and platelets get activated. Our simulations confirm clinical results where TG did not show a better outcome when compared to the CD.