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.     

Modeling the Afferent Dynamics of the Baroreflex Control System

In this study we develop a modeling framework for predicting baroreceptor firing rate as a function of blood pressure. We test models within this framework both quantitatively and qualitatively using data from rats. The models describe three components: arterial wall deformation, stimulation of mechanoreceptors located in the BR nerve-endings, and modulation of the action potential frequency. The three sub-systems are modeled individually following well-established biological principles. The first submodel, predicting arterial wall deformation, uses blood pressure as an input and outputs circumferential strain. The mechanoreceptor stimulation model, uses circumferential strain as an input, predicting receptor deformation as an output. Finally, the neural model takes receptor deformation as an input predicting the BR firing rate as an output. Our results show that nonlinear dependence of firing rate on pressure can be accounted for by taking into account the nonlinear elastic properties of the artery wall. This was observed when testing the models using multiple experiments with a single set of parameters. We find that to model the response to a square pressure stimulus, giving rise to post-excitatory depression, it is necessary to include an integrate-and-fire model, which allows the firing rate to cease when the stimulus falls below a given threshold. We show that our modeling framework in combination with sensitivity analysis and parameter estimation can be used to test and compare models. Finally, we demonstrate that our preferred model can exhibit all known dynamics and that it is advantageous to combine qualitative and quantitative analysis methods.     

基于卷积神经网络-长短期记忆神经网络模型利用光学体积描记术重建动脉血压波信号

目的 直接动脉血压（arterial blood pressure,ABP）连续监测是侵入式的,传统袖带式的间接血压测量法无法实现连续监测。既往利用光学体积描记术（photoplethysmography,PPG）实现了连续无创血压监测,但其为收缩压和舒张压的离散值,而非ABP波的连续值,本研究期望基于卷积神经网络-长短期记忆神经网络（CNN-LSTM）利用PPG信号波重建ABP波信号,实现连续无创血压监测。方法 构建CNN-LSTM混合神经网络模型,利用重症监护医学信息集（medical information mart for intensive care,MIMIC）中的PPG与ABP波同步记录信号数据,将PPG信号波经预处理降噪、归一化、滑窗分割后输入该模型,重建与之同步对应的ABP波信号。结果 使用窗口长度312的CNN-LSTM神经网络时,重建ABP值与实际ABP值间误差最小,平均绝对误差（mean absolute error,MAE）和均方根误差（root mean square error,RMSE）分别为2.79 mmHg和4.24 mmHg,余弦相似度最大,重建ABP值与实际ABP值一致性和相关性情况良好,符合美国医疗器械促进协会（Association for the Advancement of Medical Instrumentation,AAMI）标准。结论 CNN-LSTM混合神经网络可利用PPG信号波重建ABP波信号,实现连续无创血压监测。     

The role of the surrounding tissue in the propagation of waves through the arterial system.

A theoretical analysis of the flow in arteries is presented, taking into consideration the role played by the surrounding tissues in determining the speed of propagatoion and the damping of the blood pressure pulse. This study was undertaken (a) to exhibit a method of computing the flow properties with a more nearly accurate model, (b) to see if the displacement on the skin would be related to the arterial wall displacement, and hence to pressure, velocity and flow rate of blood in the artery, and if it is likely to be measurable. It was found that the pressure of the 'viscous' part in the surrounding tissue increases the pulse velocity and the damping of the wave over the values found by other models which considered only thick-walled elastic tubes with no surrounding tissue. This study also shows that measurements on the skin can provide information about changes in arterial circulation due to diseases such as: edema, arteriosclerosis and others where the Young's modulus for either the arterial wall or the surrounding tissues is altered.     

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.     

Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress

A mathematical model of the arterial baroreflex was developed and used to assess the stability of the reflex and its potential role in producing the low-frequency arterial blood pressure oscillations called Mayer waves that are commonly seen in humans and animals in response to decreased central blood volume. The model consists of an arrangement of discrete-time filters derived from published physiological studies, which is reduced to a numerical expression for the baroreflex open-loop frequency response. Model stability was assessed for two states: normal and decreased central blood volume. The state of decreased central blood volume was simulated by decreasing baroreflex parasympathetic heart rate gain and by increasing baroreflex sympathetic vaso/venomotor gains as occurs with the unloading of cardiopulmonary baroreceptors. For the normal state, the feedback system was stable by the Nyquist criterion (gain margin = 0.6), but in the hypovolemic state, the gain margin was small (0.07), and the closed-loop frequency response exhibited a sharp peak (gain of 11) at 0.07 Hz, the same frequency as that observed for arterial pressure fluctuations in a group of healthy standing subjects. These findings support the theory that stresses affecting central blood volume, including upright posture, can reduce the stability of the normally stable arterial baroreflex feedback, leading to resonance and low-frequency blood pressure waves.     

A dynamic model of angiotensin II infusion experiments

Applications of control theory in studies of biological system dynamics have come to be called compartmental modelling. A second order, nonlinear, compartmental model is developed which describes the dynamics of the hormone angiotensin II (AII) and arterial blood pressure (BP) during AII infusion experiments. The model is partially identified using dose response data for constant infusion rates between 0.01 and 0.10 μg/kg/min over a period of several minutes. This study represents a first step in understanding the dynamics of regulation of arterial blood pressure by the renin-angiotensin system. All is a vasoconstrictor and is known to participate in the natural regulation of BP. AII is also believed to be an agent in the development of hypertension and atherosclerosis. The model is used to identify causal mechanisms which are consistent both with the established correlation between plasma AII concentration and arterial BP and with current physiological knowledge. The study demonstrates how a simple state variable model can be used to provide guidance concerning the design of future infusion experiments.     

On the transmission of external pressure fluctuations to the cerebrospinal fluid

A theory has been formulated to explain the manner in which external pressure fluctuations are transmitted to the cerebrospinal fluid (CSF). The theory is based upon a three-compartment model which consists of the cerebral ventricles, the basal cisterns and spinal subarachnoid space, and the cortical subarachnoid space. The external pressure disturbance is represented by a Fourier series summed over the frequency ω. The mathematical analysis leads to a time constant τ which depends upon the compliances of the spinal region and sources of external pressure fluctuations, the rate of CSF absorption and the rate of fluid transfer between compartments. For arterial pulsations where ωτ ? 1, the theory is in accord with the experimental observations that (i) the arterial and CSF pulse waves are nearly identical in shape, and (ii) the amplitude of the CSF pulse wave increases with intracranial pressure. Moreover, it predicts that the amplitude of the wave will be larger in the spinal region than in the ventricles. The theory also accounts for the observation of one per minute pulse waves observed in hydrocephalic patients with decreased absorption rates.     

Dynamical systems analysis of arterial blood pressure signals in relation to heart rate fluctuations in chick embryos

We attempted a new approach based on a modern dynamical system theory to reconstruct the arterial blood pressure signals in relation to heart rate fluctuations of developing chick embryos. The dynamical systems approach in general is to model a phenomenon that is presented by a single time series record and approximate the dynamical property (e.g. heart rate fluctuations) of a system based only on information contained in a single-variable (arterial blood pressure) of the system. The time-series data of the arterial blood pressure was reconstructed in 3-dimensional space to draw characteristic orbits. Since the reconstructed orbits of the blood pressure should retain information contained in the pressure signals, we attempted to derive instantaneous heart rate (IHR) from the reconstructed orbits. The derived IHR presenting HR fluctuations coincided well with the IHR obtained conventionally from the peak-to-peak time intervals of the maximum blood pressure. Movements of the reconstructed orbits of the arterial blood pressure in 3-dimensional space reflected HR fluctuations (i.e. transient decelerations and accelerations).     

Baroreceptor reflex control of heart rate in rats studied by induced and autogenic changes in arterial pressure

The function of the arterial baroreflex has traditionally been assessed by measurement of reflex changes in heart rate (HR) or sympathetic nerve activity resulting from experimenter-induced manipulation of arterial blood pressure (the Oxford method, also termed the pharmacological method). However, logistical and flexibility limitations of this technique have promoted the development of new methods for assessing baroreflex function such as the evaluation of changes in spontaneous arterial pressure and HR. Although this new spontaneous method has been validated in dogs and humans, it has not been rigorously tested in rats. In the present study, the method of correlating spontaneous changes in systolic blood pressure and HR was evaluated in resting, normotensive Sprague-Dawley rats. This technique was found to be neither reliable nor valid under the conditions employed in the present protocol. We also tested a variation of the spontaneous method that evaluates particular sequences of data during which arterial pressure and pulse interval are changing in the same direction for at least three consecutive heartbeats (the sequence method). The sequence method did not provide extra reliability or validity over the spontaneous method. We conclude that due to the restricted range of variability obtained by measuring spontaneous blood pressure fluctuations, the spontaneous and sequence techniques do not provide data that are comparable to the traditional method of assessing HR changes triggered by arterial blood pressure increases and decreases induced by vasoactive drugs. However, it is possible that surgical stress obscured the relationship between blood pressure and HR, and therefore additional studies are needed to determine whether the spontaneous and sequence methods can be applied to rats during different behavioral states.     

A provisional model to predict blood pressure response to biofeedback-assisted relaxation

Blood pressure (BP) response to biofeedback-assisted relaxation is not uniform among hypertensive individuals. The purpose of this exploratory study was to determine if selected psychophysiological variables could be used to identify individuals able to lower blood pressure using biofeedback-assisted relaxation. Responders were defined using a preset criterion of 5 mm Hg or greater decrease in mean arterial pressure. A logistic regression model derived from five variables (heart rate, finger temperature, forehead muscle tension, plasma renin response to furosemide, and mean arterial pressure response to furosemide) provided significant predictive power for BP response, exhibiting a sensitivity of 84.6% and a specificity of 80.0%. With future validation, the proposed model may provide useful information to identify patients likely to benefit from biofeedback-assisted relaxation.     

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.     

Arteriovenous ratios of angiotensin II during acute infusion experiments: a model-based nalysis

The hormone angiotensin II (AII) is a vascocontrictor known to participate in the natural regulation of blood pressure via the renin-angiotensin system. A third-order model was developed which describes the dynamics of venous and arterial plasma AII concentrations (PAC) and mean arterial blood pressure (BP) during acute constant rate AII infusion experiments. The model is calibrated using approximate blood circulation rates and steady-state PAC and BP data for published experiments in sheep. Analysis of the dynamic model demonstrates that local changes in PAC during the first several minutes of acute infusion are characterized by the comparatively rapid distribution of exogenous AII making its forward passage across the blood circulation, combined with the more gradual elevation of exogenous AII recycled through the circulation. This analysis explains the observed divergence in physiological levels of venous and arterial PAC at steady state in terms of the monotonic net clearance of elevated levels of circulating AII along the circulatory path between the point of infusion and the two sites at which the PAC measurements are taken. The model suggests that the differing arteriovenous AII concentration ratios and differing PAC and BP relationships reported for different dose-response experiments may be explained in part by differences in the specific infusion and measurement sites employed in those experiments.     

Modeling water flow through arterial tissue

A simple model of, water flow through deformable porous media has been developed with emphasis on application to arterial walls. The model incorporates a strain-dependent permeability function into Darcy's Law which is coupled, to the force balance for the bulk material. A simple analytical expression relating water flux (volume flux) to pressure differential is developed which shows how strain-dependent permeability can lead to a reduction in hydraulic conductivity with increasing differential pressure as observed in experiments with arteries. The variation of permeability with position in the wall, which may influence the convective diffusion of macromolecules, is determined for both cylindrical and planar segments and a marked influence of geometry is noted.     

正常人动脉压力感受器反射开环增益的计算方法

为在闭环条件下测得正常人的动脉压力感受器反射的开环增益(G),利用三个不同特点的血压调节模型,导出了G的计算表达式及G与血液动力学变量间的关系。用所导出的公式,通过适当的实验设计,便可近似地分别计算出颈动脉窦和主动脉弓压力感受器反射的开环增益。文中给出了计算的例子。此外,还利用导出的关系式分析了心率、外周阻力的调节变化对系统开环增益的相对贡献及输入压力水平对增益效果的影响。     

清醒羊低氧性肺动脉高压及逆转中肺血管对外源性血小板活化因子的反应

本实验复制了清醒羊低氧性肺动脉高压（ＨＰＨ）及其逆转模型,动态观察了在ＨＰＨ发生、发展及逆转中肺、体循环对外源性血小板活化因子（ＰＡＦ）的反应。结果表明：（１）ＰＡＦ对清醒羊常氧期、ＨＰＨ及ＨＰＨ逆转期均为一强效应肺血管收缩剂,且呈剂量依赖性;（２）ＨＰＨ期：ＰＡＦ对肺血管加压反应增值百分离较常氧期及ＨＰＨ逆转期低;（３）ＰＡＦ对三期体动脉压、心率无明显影响,却能降低各期心输出量。     

Blood pressure and blood flow variation during postural change from sitting to standing: model development and validation.

Short-term cardiovascular responses to postural change from sitting to standing involve complex interactions between the autonomic nervous system, which regulates blood pressure, and cerebral autoregulation, which maintains cerebral perfusion. We present a mathematical model that can predict dynamic changes in beat-to-beat arterial blood pressure and middle cerebral artery blood flow velocity during postural change from sitting to standing. Our cardiovascular model utilizes 11 compartments to describe blood pressure, blood flow, compliance, and resistance in the heart and systemic circulation. To include dynamics due to the pulsatile nature of blood pressure and blood flow, resistances in the large systemic arteries are modeled using nonlinear functions of pressure. A physiologically based submodel is used to describe effects of gravity on venous blood pooling during postural change. Two types of control mechanisms are included: 1) autonomic regulation mediated by sympathetic and parasympathetic responses, which affect heart rate, cardiac contractility, resistance, and compliance, and 2) autoregulation mediated by responses to local changes in myogenic tone, metabolic demand, and CO(2) concentration, which affect cerebrovascular resistance. Finally, we formulate an inverse least-squares problem to estimate parameters and demonstrate that our mathematical model is in agreement with physiological data from a young subject during postural change from sitting to standing.     

A model of the interaction between autoregulation and neural activation in the brain

In this paper a model is proposed that predicts the response of the cerebral vasculature to changes in arterial blood pressure, arterial CO2 concentration and neural stimulation. Cerebral blood flow (CBF) is assumed to be controlled through changes in arterial compliance, and hence arterial resistance and volume, through three feedback mechanisms, which act in a linear additive manner, based on CBF, arterial CO2 and neural stimulus. Together with arterial, capillary and venous compartments, a tissue compartment is included, which contributes partly to the initial rise found in the deoxyhaemoglobin response to neural activation. Dynamic simulations of the model under different conditions show that there is significant interaction between the autoregulation and activation processes, and that the level of autoregulation has a strong influence on the CBF and deoxyhaemoglobin responses to neural activation. Overshoot in the deoxyhaemoglobin response is eliminated completely in the absence of this regulation. The feedback mechanism time constants significantly affect the CBF and deoxyhaemoglobin responses. Changes in arterial blood pressure (ABP) are found to have a strong influence on the neural activation response, with the amplitude of the response decreasing significantly at high baseline ABP. Dynamic changes in ABP also have a significant and potentially confounding impact on the measured deoxyhaemoglobin response to neural activation.     

Hemodynamic Changes in Response to Hyperacute Spinal Trauma in a Swine Model

Acute spinal cord injury (ASCI) is a devastating event that can have severe hemodynamic consequences, depending on location and severity of the lesion. Knowledge of hyperacute hemodynamic changes is important for researchers using porcine models of thoracic ASCI. The goal of this study was to determine the hyperacute hemodynamic changes observed after ASCI when using pigs as their own controls. Five Yucatan gilts were anesthetized, and a dorsal laminectomy performed at T10-T12. Standardized blunt trauma was applied for 5 consecutive min, and hemodynamic variables were collected 5 min before ASCI, and at 2, 4, 6, 8, 10, 20, 30, 60, 80 and 120 min after ASCI. Arterial blood gas samples were collected at 60 min and 10 min before, and at 30 min and between 120 and 240 min after ASCI. Parametric data were analyzed using a mixed effects model with time point as the fixed factor and subject as the random factor. We found no effect on heart rate, pulse pressure, SpO₂, EtCO₂, and respiratory rate between baseline and timepoints after ASCI. Diastolic arterial pressure, mean arterial pressure, and systolic arterial pressure fell significantly by 18%, 16%, and 15%, respectively, at 2 min after ASCI. However, none of the decrements in arterial pressures resulted in hypotension at any time point. Heart rate did not change significantly after ASCI. Blood glucose progressively increased to 50% above baseline between 120 and 240 minutes after ASCI. Low thoracic ASCI caused a consistent and statistically significant but clinically minor hyperacute decrease in arterial pressures (-15%) that did not produce hypotension or metabolic changes suggestive of tissue hypoperfusion. Our findings using this model suggest that mean arterial pressures should be maintained above 85 mm Hg prior to spinal trauma in order to avoid hypotensive states after ASCI.Abbreviations and Acronyms: ASCI, acute spinal cord injury; BE, base excess; BT, core body temperature; CRI, continuous rate infusion; DAP, diastolic arterial pressure; EtCO2, expired partial pressure of carbon dioxide; EtISO, end tidal isoflurane concentration; FLK, fentanyl lidocaine ketamine; Glu, glucose; HR, heart rate; Hb, hemoglobin; Lac, lactate; LRS, lactated Ringer’s solution; MAP, mean arterial pressure; MLK, morphine lidocaine ketamine; PaCO2, arterial partial pressure of carbon dioxide; PaO2, arterial partial pressure of oxygen; PP, pulse pressure; RR, respiratory rate; SAP, systolic arterial pressure; SCI, spinal cord injury