Studies on the relations between pressure and flow in the circulation

Simple theoretical models are proposed for the study of the interdependence between cardiac contraction, arterial pressure, and capillary drainage. The relation between pressure and flow is derived for a model of branching distensible tubes taking into account the finite pulse wave velocity. Equations are derived both for the case where the pulse wave is non-distorted and for the case where the wave is damped and distorted to a limited extent. Following the model of J. W. Remington and W. F. Hamilton (1947), the former case is applied to the larger arteries. Expressions are developed for the stroke volume, cardiac ejection, and systolic arterial storage in both the steady and non-steady states. Expressions for the percentage discrepancy involved in the computation of these quantities from a single tube model as contrasted with a multi-branched model are derived. For typical cases these discrepancies are small and thus credence is lent to the further use of the simpler single tube model which requires fewer independent parameters. It is also shown that the formulae for stroke volume and arterial storage are only slightly sensitive to changes in pulse wave velocities, and that for some purposes it would seem permissible to assume an infinite velocity. The problem of capillary drainage is discussed, and the consequences of equations developed for the case of a distorted wave are shown to compare favorably with published experimental data. An approximate boundary condition for capillary drainage is derived. Finally, A. V. Hill's velocity load equation for muscle is used to obtain a first approximation for the velocity of cardiac contraction in terms of the initial arterial pressure, the heart radius, and the parameters of the heart musculature. It is shown how methods developed for stroke volume determination from the pressure contour may be used to estimate the heart and “air chamber” parameters. Use of these parameters and those obtained by other independent measurements permits the principle variables to be determined numerically.     

Effects of arterial wall models and measurement uncertainties on cardiovascular model predictions

We developed a methodology to assess and compare the prediction quality of cardiovascular models for patient-specific simulations calibrated with uncertainty-hampered measurements. The methodology was applied in a one-dimensional blood flow model to estimate the impact of measurement uncertainty in wall model parameters on the predictions of pressure and flow in an arterial network. We assessed the prediction quality of three wall models that have been widely used in one-dimensional blood flow simulations. A 37-artery network, previously used in one experimental and several simulation studies, was adapted to patient-specific conditions with a set of three clinically measurable inputs: carotid–femoral wave speed, mean arterial pressure and area in the brachial artery. We quantified the uncertainty of the predicted pressure and flow waves in eight locations in the network and assessed the sensitivity of the model prediction with respect to the measurements of wave speed, pressure and cross-sectional area. Furthermore, we developed novel time-averaged sensitivity indices to assess the contribution of model parameters to the uncertainty of time-varying quantities (e.g., pressure and flow). The results from our patient-specific network model demonstrated that our novel indices allowed for a more accurate sensitivity analysis of time-varying quantities compared to conventional Sobol sensitivity indices.     

In vitro determination of the pressure-diameter relationship and velocity profiles by ultrasonic technics. In vivo application

A good knowledge of arterial flow mechanics and of the phenomena associated with fluid-boundary interactions is necessary for the determination of some fundamental parameters such as velocity, pressure and pressure-diameter relationship during a cardiac cycle. Ultrasonic techniques were developed on a test bench and directly applied to animals without major modification. On such a test bench allowing a good simulation of physiological type flows, velocity field and pressure-diameter relationship were determined. In vivo application of these techniques allowed a systematic analysis of velocity profiles in the rabbit abdominal aorta and a precise approach of rheological properties of the vascular wall.     

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.     

Computer simulation of arterial flow with applications to arterial and aortic stenoses

A computer model for simulating pressure and flow propagation in the human arterial system is developed. The model is based on the one-dimensional flow equations and includes nonlinearities arising from geometry and material properties. Fifty-five arterial segments, representing the various major arteries, are combined to form the model of the arterial system. Particular attention is paid to the development of peripheral pressure and flow pulses under normal flow conditions and under conditions of arterial and aortic stenoses. Results show that the presence of severe arterial stenoses significantly affects the nature of the distal pressure and flow pulses. Aortic stenoses also have a profound effect on central and peripheral pressure pulse formation. Comparison with the published experimental data suggests that the model is capable of simulating arterial flow under normal flow conditions as well as conditions of stenotic obstructions in a satisfactory manner.     

Difficulties in noninvasive cardiac stroke volume determination in "small" blood circulation systems]

Pulsewave velocity analysis is an option for the noninvasive determination of cardiac output and the evaluation of additional haemodynamic parameters. An algorithm we developed ourselves has been established as a method for the measurement of cardiac output in rabbits. The effectiveness of this program was investigated by monitoring controlled hypoxia-induced alterations in the circulatory system. Calculated values were compared with direct measurements of cardiac output with a Doppler flow probe placed in the ascending aorta. Within the physiological framework of blood gas analysis, a good correlation was found between the two methods. In the case of hypoxia-induced depression, however, the two methods showed diverging results, presumably due to arrhythmia-induced wave reflections within the arterial vascular tree. This makes an extension of the algorithm necessary to take account for these case.     

A computer-assisted noninvasive monitoring system with graphic display of online determination of cardiovascular parameters

During anesthesia the cardiovascular system is usually assessed on the basis of heart rate and arterial pressure, although the most important hemodynamic measurement is that of flow. A method for the non-invasive measurement of cardiac output is based on thoracic electrical bio-impedance. The NCCOM3-R7 is a non-invasive cardiac output monitor that makes use of thoracic electrical bioimpedance, which has been shown to provide results comparable with thermodilution in various hemodynamic states both in animals and humans. A new on-line hemodynamic monitoring system has been developed using the non-invasive NCCOM3-R7 (BoMed) cardiac output monitor, a portable microcomputer (NEC Multispeed) in connection with a software package CDDP-1 (BoMed), a Dinamap automatic arterial pressure monitor (Critikon) and an additional 14" display. Every 16 heart beats the cardiac output monitor transfers 11 cardiodynamic parameters in ASCII-format to the microcomputer, where the data are stored. Using the CDDP-1 program the current cardiodynamic status of the patient is displayed numerically and graphically on the monitor screen. Mean arterial pressure is determined by Dinamap and the data must be entered manually in the menu. The program then calculates systemic vascular resistance and left ventricular work index, the CVP being set to 3 torr and PAOP to 6 torr. In the graphic display the current hemodynamic status is shown and the underlying situation is analyzed in terms of systemic vascular resistance and volume-dependent contractility. The reliability of this on-line monitoring system is demonstrated in a high-risk patient.(ABSTRACT TRUNCATED AT 250 WORDS)     

Normal hemodynamic parameters in conscious Wistar rats

The evolution of different hemodynamic parameters with ponderal growth has been studied in conscious Wistar rats. The thermodilution method has been used to determine cardiac output and related variables. The results suggest that, between animal weight and the different hemodynamic parameters, there is a direct proportional relationship to blood volume, mean arterial pressure, cardiac output, stroke volume and total peripheral resistance, and an indirect proportional relationship to heart rate, cardiac index and stroke volume index. Body weight, therefore, plays a major role in hemodynamic determination, this having to be kept in mind when designing the experiment.     

Effect of portacaval surgical anastomosis on systemic and splanchnic hemodynamics in portal hypertensive, cirrhotic rats

The effect of surgical end-to-side portacaval anastomosis (PCSA) on systemic and splanchnic circulation has been studied in cirrhotic rats with portal hypertension (CCl4-phenobarbital method) and in control animals. Hemodynamics have been measured using the microsphere technique, with a reference sample for the systemic hemodynamic measurements, and intrasplenic injection for portal systemic shunting rate measurements. Compared with controls, sham-operated (SO) cirrhotic rats showed a hyperdynamic circulation with increased cardiac output (CO) and decreased mean arterial pressure and peripheral resistances. PCSA in control rats induced only a small change in systemic hemodynamics, with parallel decreases in arterial pressure and peripheral resistances, and a small, nonsignificant increase in CO. In cirrhotic rats, PCSA induced a decrease of CO to values similar to those of control rats, with an increase in total peripheral resistances. PCSA induced an increase in hepatic arterial blood flow in control and in cirrhotic rats, portal pressure becoming in this latter group not different from that of control rats. Blood flow to splanchnic organs was higher in SO cirrhotic than in SO control animals. Thus portal venous inflow was also increased in SO cirrhotic rats. PCSA induced an increase in portal venous inflow in control rats, which was only significant in cirrhotic rats when expressed as a percentage of CO. In SO control animals, a significant correlation was observed between total peripheral resistances and splanchnic arteriolar resistances and between CO and splanchnic blood flow. These correlations were not observed in cirrhotic rats. These results do not support the hypothesis that hyperdynamic circulation shown by cirrhotic rats is based on increases in splanchnic blood flow and (or) massive portal systemic shunting.     

A thick walled viscoelastic model for the mechanics of arteries

A knowledge of the mechanics of arteries is of importance in the determination of vessel rheological properties and in the studies of blood flow and certain arterial diseases. Most existing arterial models treat only wave motions; however, other types of motion, in particular those associated with flow development and other end effects, occur in the vascular system. Thus, a model is needed which can be applied to a variety of possible types of motion.

An arterial model is described which includes the effects of thick walls, linear viscoelasticity, and wall tethering. The forms of the displacements and stresses are found independently of the exact form of the applied fluid stresses; thus, the results are applicable to a range of possible dynamical conditions. Displacements and stress states can then be found from experimental or theoretical knowledge of the blood pressure and flow. The results are applied to flow development and wave propagation regions in the arteries.     

Arterial pressure contour analysis for estimating human vascular properties.

The form of an arterial blood pressure curve during the diastolic portion of the cardiac cycle was here employed to identify parameters in a third-order model of the vascular system. Calculated elastic and intertial characteristics of this fitted model then became clinically accessible indices of corresponding real vascular properties. This technique incurred no risk and little discomfort for the patient. Tested in theory, in animal experimentation, and in human observations, our procedure utilized a Gauss-Newton algorithm via digital computer to provide rapid model solutions from different starting values, from multiple measurements sites, and from normal or diseased patients. Model parameters thus determined defined ranges of normal variation and suggested a less compliant arterial bed in hypertensive than in normotensive patients.     

Computer simulation of blood flow in the human arm

This paper considers a finite element method to characterize blood flow in the human arm arteries. A set of different pressure waveforms, which represent normal and diseased heart pulses, is used for the proximal boundary conditions, and a modified Windkessel model is used for the distal arterial boundary conditions. A comparison of the distal pressure and flow waveforms, for each different proximal pressure, is made to determine whether such waveforms are significantly altered from normal waveforms. The results show that the distal pressure and/or flow waveforms in certain cases are sufficiently different to be possibly used as a diagnostic indicator of an abnormal heart condition. Also considered is the effect of stenosis, change of compliance, and dilatation of the distal beds on the pressure and flow waveforms. A stenosis which has an area reduction of greater than approximately 75% is found to significantly alter both the distal pressure and flow waveforms. Changes in arterial compliance, however, do not strongly influence the waveforms. Dilatation of distal vascular beds is simulated by reducing the lumped resistance of these beds, and this reduction increases mean flow and decreases mean distal pressure, but has little effect on the basic shape of either the pressure or flow waveform.     

Anatomically based finite element models of the human pulmonary arterial and venous trees including supernumerary vessels.

Studies of the origin of pulmonary blood flow heterogeneity have highlighted the significant role of vessel branching structure on flow distribution. To enable more detailed investigation of structure-function relationships in the pulmonary circulation, an anatomically based finite element model of the arterial and venous networks has been developed to more accurately reflect the geometry found in vivo. Geometric models of the arterial and venous tree structures are created using a combination of multidetector row X-ray computed tomography imaging to define around 2,500 vessels from each tree, a volume-filling branching algorithm to generate the remaining accompanying conducting vessels, and an empirically based algorithm to generate the supernumerary vessel geometry. The explicit generation of supernumerary vessels is a unique feature of the computational model. Analysis of branching properties and geometric parameters demonstrates close correlation between the model geometry and anatomical measures of human pulmonary blood vessels. A total of 12 Strahler orders for the arterial system and 10 Strahler orders for the venous system are generated, down to the equivalent level of the terminal bronchioles in the bronchial tree. A simple Poiseuille flow solution, assuming rigid vessels, is obtained within the arterial geometry of the left lung, demonstrating a large amount of heterogeneity in the flow distribution, especially with inclusion of supernumerary vessels. This model has been constructed to accurately represent available morphometric data derived from the complex asymmetric branching structure of the human pulmonary vasculature in a form that will be suitable for application in functional simulations.     

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.     

A global mathematical model of the cerebral circulation in man

A mathematical model of the cerebral circulation has been formulated. It was based on non-linear equations of pulsatile fluid flow in distensible conduits and applied to a network simulating the entire cerebral vasculature, from the carotid and vertebral arteries to the sinuses and the jugular veins. The quasilinear hyperbolic system of equations was numerically solved using the two-step Lax-Wendroff scheme. The model's results were in good agreement with pressure and flow data recorded in humans during rest. The model was also applied to the study of autoregulation during arterial hypotension. A close relationship between cerebral blood flow (CBF) and capillary pressure was obtained. At arterial pressure of 80 mmHg, the vasodilation of the pial arteries was unable to maintain CBF at its control value. At the lower limit of autoregulation (60 mm Hg), CBF was maintained with a 25% increase of zero transmural pressure diameter of nearly the whole arterial network.     

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.     

Personalized Computer Simulation of Diastolic Function in Heart Failure

The search for a parameter representing left ventricular relaxation from non-invasive and invasive diagnostic tools has been extensive, since heart failure (HF) with preserved ejection fraction (HF-pEF) is a global health problem. We explore here the feasibility using patient-specific cardiac computer modeling to capture diastolic parameters in patients suffering from different degrees of systolic HF. Fifty eight patients with idiopathic dilated cardiomyopathy have undergone thorough clinical evaluation, including cardiac magnetic resonance imaging (MRI), heart catheterization, echocardiography, and cardiac biomarker assessment. A previously-introduced framework for creating multi-scale patient-specific cardiac models has been applied on all these patients. Novel parameters, such as global stiffness factor and maximum left ventricular active stress, representing cardiac active and passive tissue properties have been computed for all patients. Invasive pressure measurements from heart catheterization were then used to evaluate ventricular relaxation using the time constant of isovolumic relaxation Tau (s). Parameters from heart catheterization and the multi-scale model have been evaluated and compared to patient clinical presentation. The model parameter global stiffness factor, representing diastolic passive tissue properties, is correlated signif-icantly across the patient population with s. This study shows that multi-modal cardiac models can successfully capture diastolic (dys) function, a prerequisite for future clinical trials on HF-pEF.     

Application of 1D blood flow models of the human arterial network to differential pressure predictions

A new application of 1D models of the human arterial network is proposed. We take advantage of the sensitivity of the models predictions for the pressure profiles within the main aorta to key model parameter values. We propose to use the patterns in the predicted differences from a base case as a way to infer to the most probable changes in the parameter values. We demonstrate this application using an impedance model that we have recently developed (Johnson, 2010). The input model parameters are all physiologically related, such as the geometric dimensions of large arteries, various blood properties, vessel elasticity, etc. and can therefore be patient specific. As a base case, nominal values from the literature are used. The necessary information to characterize the smaller arteries, arterioles, and capillaries is taken from a physical scaling model (West, 1999). Model predictions for the effective impedance of the human arterial system closely agree with experimental data available in the literature. The predictions for the pressure wave development along the main arteries are also found in qualitative agreement with previous published results. The model has been further validated against our own measured pressure data in the carotid and radial arteries, obtained from healthy individuals. Upon changes in the value of key model parameters, we show that the differences seen in the pressure profiles correspond to qualitatively different patterns for different parameters. This suggests the possibility of using the model in interpreting multiple pressure data of healthy/diseased individuals.     

Large signal,low frequency characteristics of the systemic arterial system

A 4-parameter model to represent the relationship between slowly changing arterial inflow and the resulting pressure change was proposed. In order to determine values for the parameters and test the model in a wide range of pressure and flow, two experimental procedures were used. Firstly, slow oscillations in aortic pressure (period, 4 sec) were produced by means of an external pump connected to the aorta, and secondly, cardiac arrest was induced for about 2–3 sec by stimulating the right vagus. It was shown mathematically that the pressure and pump displacement information obtained in these experiments could be related to the model parameters. Values for the model parameters were obtained in 6 anesthetized dogs using a least squares adjustment procedure. Using aX ²-test, it was concluded that the model represented the arterial system for each animal well.