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.     

On the propagation of a wave front in viscoelastic arteries

In formulating a mathematical model of the arterial system, the one-dimensional flow approximation yields realistic pressure and flow pulses in the proximal as well as in the distal regions of a simulated arterial conduit, provided that the viscoelastic damping induced by the vessel wall is properly taken into account. Models which are based on a purely elastic formulation of the arterial wall properties are known to produce shocklike transitions in the propagating pulses which are not observed in man under physiological conditions. The viscoelastic damping characteristics are such that they are expected to reduce the tendency of shock formation in the model. In order to analyze this phenomenon, the propagation of first and second-order pressure waves is calculated with the aid of a wave front expansion, and criteria for the formation of shocks are derived. The application of the results to the human arterial system show that shock waves are not to be expected under normal conditions, while in case of a pathologically increased pressure rise at the root of the aorta, shocklike transitions may develop in the periphery. In particular, it is shown that second-order waves never lead to shock formation in finite time for the class of initial conditions and mechanical wave guides which are of interest in the mammalian circulation.     

Steady flow of a viscous fluid through a network of tubes with applications to the human arterial system

The paper presents a finite-element model for the analysis of steady flow of a viscous fluid through a connected system of elastic tubes with the aim of simulating the conditions of blood flow through the human arterial system. The governing equations of the model are non-linear in character and are solved through an iterative computational procedure. This model is capable of incorporating the effects of stenosis on flow and pressure. Typical results are presented and discussed. Quantitative results have been obtained on blood flow through a model of the human arterial system corresponding to the sets of prescribed conditions at the terminations. Also computational results on the effect of stenosis in typical arteries of the system are presented.     

Low Reynolds number turbulence modeling of blood flow in arterial stenoses.

Moderate and severe arterial stenoses can produce highly disturbed flow regions with transitional and or turbulent flow characteristics. Neither laminar flow modeling nor standard two-equation models such as the kappa-epsilon turbulence ones are suitable for this kind of blood flow. In order to analyze the transitional or turbulent flow distal to an arterial stenosis, authors of this study have used the Wilcox low-Re turbulence model. Flow simulations were carried out on stenoses with 50, 75 and 86% reductions in cross-sectional area over a range of physiologically relevant Reynolds numbers. The results obtained with this low-Re turbulence model were compared with experimental measurements and with the results obtained by the standard kappa-epsilon model in terms of velocity profile, vortex length, wall shear stress, wall static pressure, and turbulence intensity. The comparisons show that results predicted by the low-Re model are in good agreement with the experimental measurements. This model accurately predicts the critical Reynolds number at which blood flow becomes transitional or turbulent distal an arterial stenosis. Most interestingly, over the Re range of laminar flow, the vortex length calculated with the low-Re model also closely matches the vortex length predicted by laminar flow modeling. In conclusion, the study strongly suggests that the proposed model is suitable for blood flow studies in certain areas of the arterial tree where both laminar and transitional/turbulent flows coexist.     

Frequency analysis of arterial pulse transmission in the cranial cavity]

The changes of intracranial and arterial pulse shape under functional loads (hypervolemia and intracranial hypertension) were compared. The logarithmic amplitude-frequency characteristics were found and used for the synthesis of equivalent electrical circuit of arterial pressure pulses transmission on cerebrospinal fluid (CSF) in the cranial cavity. The model obtained points to the necessity of taking into account the induction which was not performed in the earlier models of the CSF-system. It is found that the attenuation factor permitted to estimate the stability of the intracranial circulation system to input influences under different functional conditions.     

Small disturbance in arterial blood flow

The behaviour of a small disturbance in an arterial blood flow has been studied analytically. The growth equation governing growth or decay of a disturbance has been obtained and solved. The behaviour of wave amplitude has been investigated as the wave propagates in time. The application of results to the human arterial system shows that the shock waves are not expected under normal physiological conditions. In the case of a pathologically increased pressure rise at the root of aorta, shock-like transitions may develop in the periphery. It is observed that the friction effects are to resist the tendency of shock formation in arteries.     

Compliance characteristics of the hind-limb arterial system of normal and hypertensive rabbits

Pressure transients resulting from square-wave changes in abdominal aortic blood flow rate were used to derive effective arterial compliance and peripheral resistance of the hind-limb circulation of anaesthetized rabbits. The model for deriving these parameters proved applicable if step changes in flow were kept less than 35% of mean flow. Under resting conditions, the effective hind-limb arterial compliance of normal rabbits averaged 3.46 X 10(-3) mL/mmHg (1 mmHg = 133.322 Pa). Hind-limb arterial compliance decreased with increasing pressure at low arterial pressures, but unlike compliance of isolated arterial segments, compliance did not vary at and above normal resting pressures. Baroreflex destimulation (bilateral carotid artery occlusion) caused an increase in effective hind-limb vascular resistance at 48.4% and a decrease of arterial compliance of 50.7%, so that the constant for flow-induced arterial pressure changes (resistance times compliance) was largely unchanged. Similarly, the arterial time constant for rabbits with chronic hypertension was similar to that for controls because threefold increases in hind-limb vascular resistance were offset by decreases in compliance. Reflex-induced decreases in arterial compliance are probably mediated by sympathetic nerves, whereas decreases associated with hypertension are related to wall hypertrophy in conjunction with increased vasomotor tone. Arterial compliance decreased with increasing pressure in hypertensive animals, but this effect was less pronounced than in normotensive rabbits.     

An intergral method for the analysis of flow in arterial stenoses

An approximate solution is presented to the problem of incompressible flow through an axisymmetric constriction. The geometry is intended to simulate an arterial stenosis, and the solution is applicable to both mild and severe stenoses for Reynolds numbers below transition. Theoretical results obtained for specific geometries are given for the velocity distribution, pressure drop, wall shearing stress, and separation phenomena. These results reveal the significant alterations in flow caused by a stenosis. Experiments using model stenoses are described and compared with the theoretical results. Theoretical predictions of pressure drop and separation characteristics are in reasonably good agreement with the experimental observations.     

Venous occlusion plethysmography reduces arterial diameter and flow velocity

The measurement of peripheral blood flow by plethysmography assumes that the cuff pressure required for venous occlusion does not decrease arterial inflow. However, studies in five normal subjects suggested that calf blood flow measured with a plethysmograph was less than arterial inflow calculated from Doppler velocity measurements. We hypothesized that the pressure required for venous occlusion may have decreased arterial velocity. Further studies revealed that systolic diameter of the superficial femoral artery under a thigh cuff decreased from 7.7 +/- 0.4 to 5.6 +/- 0.7 mm (P less than 0.05) when the inflation pressure was increased from 0 to 40 mmHg. Cuff inflation to 40 mmHg also reduced mean velocity 38% in the common femoral artery and 47% in the popliteal artery. Inflation of a cuff on the arm reduced mean velocity in the radial artery 22% at 20 mmHg, 26% at 40 mmHg, and 33% at 60 mmHg. We conclude that inflation of a cuff on an extremity to low pressures for venous occlusion also caused a reduction in arterial diameter and flow velocity.     

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.     

A geometry-based model for non-invasive estimation of pressure gradients over iliac artery stenoses

The aim of this study was to develop and verify a model that provides an accurate estimation of the trans-lesion hyperemic pressure gradient in iliac artery stenoses in seconds by only using patient-specific geometric properties obtained from 3-dimensional rotational angiography (3DRA).Twenty-one patients with symptomatic peripheral arterial disease (PAD), iliac artery stenoses and an ultrasound based peak systolic velocity ratio between 2.5 and 5.0 underwent 3DRA and intra-arterial pressure measurements under hyperemic conditions. For each lesion, geometric properties were extracted from the 3DRA images using quantitative vascular analysis software. Hyperemic blood flow was estimated based on stenosis geometry using an empirical relation. The geometrical properties and hyperemic flow were used to estimate the pressure gradient by means of the geometry-based model. The predicted pressure gradients were compared with in vivo measured intra-arterial pressure measurements performed under hyperemic conditions.The developed geometry-based model showed good agreement with the measured hyperemic pressure gradients resulting in a concordance correlation coefficient of 0.86. The mean bias ± 2SD between the geometry-based model and in vivo measurements was comparable to results found by evaluating the actual computational fluid dynamics model (−1.0 ± 14.7 mmHg vs −0.9 ± 12.7 mmHg).The developed model estimates the trans-lesional pressure gradient in seconds without the need for an additional computational fluid dynamics software package. The results justify further study to assess the potential use of a geometry-based model approach to estimate pressure gradient on non-invasive CTA or MRA, thereby reducing the need for diagnostic angiography in patients suffering from PAD.     

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.     

Nonlinear pulse wave reflection at an arterial stenosis

A simple model is presented to analyze the effect of stenoses of different severities in a long elastic tube or artery on the pressure and flow-rate wave forms incident upon them. Wave propagation in the undisturbed tube is taken to be linear; nonlinearity arises from the quadratic dependence of stenosis pressure drop on flow rate. Before the model can be applied in practice, important physiological questions must be answered; e.g.: (a) Can the incident wave form and mean proximal pressure be regarded as given input? (b) is the mean flow rate given, or does the peripheral resistance remain constant? Results are given on the assumption that the answer to (a) is yes. The principal conclusion is that the input impedance spectrum of a stenosed artery depends strongly on the incident wave form, as well as on the severity of the stenosis and on the distance from it at which measurements are made. There is good qualitative agreement with the results of experiments and of other models.     

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.     

Pulse Decomposition Analysis of the digital arterial pulse during hemorrhage simulation

Background

Markers of temporal changes in central blood volume are required to non-invasively detect hemorrhage and the onset of hemorrhagic shock. Recent work suggests that pulse pressure may be such a marker. A new approach to tracking blood pressure, and pulse pressure specifically is presented that is based on a new form of pulse pressure wave analysis called Pulse Decomposition Analysis (PDA). The premise of the PDA model is that the peripheral arterial pressure pulse is a superposition of five individual component pressure pulses, the first of which is due to the left ventricular ejection from the heart while the remaining component pressure pulses are reflections and re-reflections that originate from only two reflection sites within the central arteries. The hypothesis examined here is that the PDA parameter T13, the timing delay between the first and third component pulses, correlates with pulse pressure. T13 was monitored along with blood pressure, as determined by an automatic cuff and another continuous blood pressure monitor, during the course of lower body negative pressure (LBNP) sessions involving four stages, -15 mmHg, -30 mmHg, -45 mmHg, and -60 mmHg, in fifteen subjects (average age: 24.4 years, SD: 3.0 years; average height: 168.6 cm, SD: 8.0 cm; average weight: 64.0 kg, SD: 9.1 kg).

Results

Statistically significant correlations between T13 and pulse pressure as well as the ability of T13 to resolve the effects of different LBNP stages were established. Experimental T13 values were compared with predictions of the PDA model. These interventions resulted in pulse pressure changes of up to 7.8 mmHg (SE = 3.49 mmHg) as determined by the automatic cuff. Corresponding changes in T13 were a shortening by -72 milliseconds (SE = 4.17 milliseconds). In contrast to the other two methodologies, T13 was able to resolve the effects of the two least negative pressure stages with significance set at p < 0.01.

Conclusions

The agreement of observations and measurements provides a preliminary validation of the PDA model regarding the origin of the arterial pressure pulse reflections. The proposed physical picture of the PDA model is attractive because it identifies the contributions of distinct reflecting arterial tree components to the peripheral pressure pulse envelope. Since the importance of arterial pressure reflections to cardiovascular health is well known, the PDA pulse analysis could provide, beyond the tracking of blood pressure, an assessment tool of those reflections as well as the health of the sites that give rise to them.     

Analysis of blood flow in the entire coronary arterial tree

A hemodynamic analysis of coronary blood flow must be based on the measured branching pattern and vascular geometry of the coronary vasculature. We recently developed a computer reconstruction of the entire coronary arterial tree of the porcine heart based on previously measured morphometric data. In the present study, we carried out an analysis of blood flow distribution through a network of millions of vessels that includes the entire coronary arterial tree down to the first capillary branch. The pressure and flow are computed throughout the coronary arterial tree based on conservation of mass and momentum and appropriate pressure boundary conditions. We found a power law relationship between the diameter and flow of each vessel branch. The exponent is approximately 2.2, which deviates from Murray's prediction of 3.0. Furthermore, we found the total arterial equivalent resistance to be 0.93, 0.77, and 1.28 mmHg.ml(-1).s(-1).g(-1) for the right coronary artery, left anterior descending coronary artery, and left circumflex artery, respectively. The significance of the present study is that it yields a predictive model that incorporates some of the factors controlling coronary blood flow. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion.     

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.     

Sustained high-pressure in the spinal subarachnoid space while arterial expansion is low may be linked to syrinx development

Syringomyelia (a spinal cord cyst) usually develops as a result of conditions that cause cerebrospinal fluid (CSF) obstruction. The mechanism of syrinx formation and enlargement remains unclear, though previous studies suggest that the fluid enters via the perivascular spaces (PVS) of the penetrating arteries of the spinal cord, and that alterations in the CSF pulse timing and pressure could contribute to enhanced PVS inflow. This study uses an idealised computational model of the PVS to investigate the factors that influence peri-arterial fluid flow. First, we used three sample patient-specific models to explore whether changes in subarachnoid space (SAS) pressures in individuals with and without syringomyelia could influence PVS inflow. Second we conducted a parametric study to determine how features of the CSF pulse altered perivascular fluid, including alterations to timing and magnitude of the peak SAS pressure, the timing of reversal from high to low pressure (diastolic phase), and the area under the pressure–time curve. The model for the patient with syringomyelia had higher net CSF inflow to the PVS than the two subjects without syringomyelia. In the parametric study, only increasing the area under the high pressure region of the SAS pulse substantially increased PVS inflow, when coupled with a temporal shift in arterial and SAS pulses. This suggests that a period of sustained high SAS pressure while arterial diameter is low may increase net CSF pumping into the PVS.     

Increases in intramuscular pressure raise arterial blood pressure during dynamic exercise.

This investigation was designed to determine the role of intramuscular pressure-sensitive mechanoreceptors and chemically sensitive metaboreceptors in affecting the blood pressure response to dynamic exercise in humans. Sixteen subjects performed incremental (20 W/min) cycle exercise to fatigue under four conditions: control, exercise with thigh cuff occlusion of 90 Torr (Cuff occlusion), exercise with lower body positive pressure (LBPP) of 45 Torr, and a combination of thigh cuff occlusion and LBPP (combination). Indexes of central command (heart rate, oxygen uptake, ratings of perceived exertion, and electromyographic activity), cardiac output, stroke volume, and total peripheral resistance were not significantly different between the four conditions. Mechanical stimulation during LBPP and combination conditions resulted in significant elevations in intramuscular pressure and mean arterial pressure from control at rest and throughout the incremental exercise protocol (P < 0.05). Conversely, there existed no significant changes in mean arterial pressure when the metaboreflex was stimulated by cuff occlusion. These findings suggest that under normal conditions the mechanoreflex is tonically active and is the primary mediator of exercise pressor reflex-induced alterations in arterial blood pressure during submaximal dynamic exercise in humans.