Pseudoaneurysm - A Late and Rare Occurrence Following Shoulder Arthroscopy

Introduction

The incidence of vascular injuries from arthroscopic surgeries has been reported to be 0.005%. Pseudoaneurysms account for 11% of those injuries.

Case Report

In this case report, we discuss a 76-year-old female who presented with a pulsatile swelling in the right shoulder after 10 years following arthroscopic rotator cuff repair. Imaging confirmed the diagnosis of a posterior circumflex artery pseudoaneurysm. The patient was successfully embolised using a transradial approach with thrombosis of the pseudoaneurysm.

Conclusion

Vascular injuries following arthroscopic shoulder surgery are rare. However, a pseudoaneurysm should be considered in patients who present with swelling at the surgical site, regardless of the post-operative interval.

Future implications of COVID-19 through Mathematical modeling

COVID-19 is a pandemic respiratory illness. The disease spreads from human to human and is caused by a novel coronavirus SARS-CoV-2. In this study, we formulate a mathematical model of COVID-19 and discuss the disease free state and endemic equilibrium of the model. Based on the sensitivity indexes of the parameters, control strategies are designed. The strategies reduce the densities of the infected classes but do not satisfy the criteria/threshold condition of the global stability of disease free equilibrium. On the other hand, the endemic equilibrium of the disease is globally asymptotically stable. Therefore it is concluded that the disease cannot be eradicated with present resources and the human population needs to learn how to live with corona. For validation of the results, numerical simulations are obtained using fourth order Runge-Kutta method.

Sudden perceived absence of foetal movement - a unique presentation of a vertebral haemangioma in pregnancy

OBJECTIVE

Symptomatic vertebral haemangiomas that present during pregnancy are extremely uncommon with few cases reported in literature. Epidural haemangiomas are rarer still with few documented.

METHODS

In this report, we describe the case of a 22-year-old pregnant patient who presented with apparent loss of foetal movement at 38 weeks' gestation. Clinical review demonstrated the foetus was well but neurological examination revealed lower limb paresthesia, paresis and evident uterine hypoesthesia. An MRI scan illustrated a haemangioma in the T1 vertebral body with an epidural component causing cord compression.

RESULTS

The management of spinal haemangiomas that present during pregnancy is a complex clinical scenario, which requires careful multidisciplinary consideration to determine if surgical intervention is appropriate. In this case, the patient had an emergency caesarean section followed by posterior decompression and laminectomy of the T1 vertebra with excellent post-operative recovery.

CONCLUSION

Gestational increase in the size of vertebral haemangiomas is well documented. We discuss a rare case in which a young pregnant patient presents with an atypical symptom of a vertebral haemangioma (uterine hypoesthesia). This case highlights the importance of prompt imaging in these scenarios and a cohesive multidisciplinary approach in order to provide optimal treatment for the patient.

Threshold conditions for global stability of disease free state of COVID-19

This article focus the elimination and control of the infection caused by COVID-19. Mathematical model of the disease is formulated. With help of sensitivity analysis of the reproduction number the most sensitive parameters regarding transmission of infection are found. Consequently strategies for the control of infection are proposed. Threshold condition for global stability of the disease free state is investigated. Finally, using Matlab numerical simulations are produced for validation of theocratical results.

Threshold condition and non pharmaceutical interventions's control strategies for elimination of COVID-19

In this work we focus on the eradication of the COVID-19 infection with the help of almost Non Pharmaceutical Interventions(NPIs), using mathematical modelling. First the basic reproduction number R 0 is investigated. Then, on the basis of sensitivity test of R 0 , the most active/sensitive parameters are presented in detail. Non Pharmaceutical Interventions(NPIs) are applied to control the sensitive parameters. The major NPIs are, stay home (isolation), sanitizers (wash hands), Treatment of side effects of infection, like throat infection etc and face mask. These NPIs helps in mitigation and reducing the size of outbreak of the disease. Threshold condition for global stability of the disease free state is investigated.The NPI's are used in different ratios to formulate a strategy. The results of these strategies are validated using Matlab software.

A model of blood supply to the brain via the carotid arteries: Effects of obstructive vs. sclerotic changes

The carotid artery is one of the major supply routes of blood to the brain and a common site of vascular disease. Obstructive and sclerotic disorders within the carotid artery impact local blood flow patterns as well as overall impedance and blood supply to the brain. A lumped parameter model and an experimental in-vitro flow loop were used to study the effects of local stenosis and stiffness in the carotid artery based on a family of phantoms with different degrees of stenosis and compliance. The model also allows independent examination of the effects of downstream resistance and compliance. Mild to moderate stenosis was found to lead to minimal (∼1%) reduction in blood supply to the brain. Reduction in mean internal carotid artery (ICA) flow was statistically significant (p< 0.01) only above 70% stenosis. On the other hand, a three-fold increase in stiffness of the carotid artery, as might occur in aging, was found to lead to a modest yet statistically significant reduction (p< 0.01) in mean ICA flow. Effects of changing downstream resistance and compliance were examined. For a given pressure waveform, reduction in downstream compliance led to altered waveform shape and reduction in peak systolic flow rates where the mean flow rates were not altered. Increased downstream resistance resulted in drastic reduction in mean flow rates.

Age structure is critical to the population dynamics and survival of honeybee colonies

Age structure is an important feature of the division of labour within honeybee colonies, but its effects on colony dynamics have rarely been explored. We present a model of a honeybee colony that incorporates this key feature, and use this model to explore the effects of both winter and disease on the fate of the colony. The model offers a novel explanation for the frequently observed phenomenon of 'spring dwindle', which emerges as a natural consequence of the age-structured dynamics. Furthermore, the results indicate that a model taking age structure into account markedly affects the predicted timing and severity of disease within a bee colony. The timing of the onset of disease with respect to the changing seasons may also have a substantial impact on the fate of a honeybee colony. Finally, simulations predict that an infection may persist in a honeybee colony over several years, with effects that compound over time. Thus, the ultimate collapse of the colony may be the result of events several years past.

Sensitivity Analysis and Optimal Control of Anthroponotic Cutaneous Leishmania

This paper is focused on the transmission dynamics and optimal control of Anthroponotic Cutaneous Leishmania. The threshold condition R0 for initial transmission of infection is obtained by next generation method. Biological sense of the threshold condition is investigated and discussed in detail. The sensitivity analysis of the reproduction number is presented and the most sensitive parameters are high lighted. On the basis of sensitivity analysis, some control strategies are introduced in the model. These strategies positively reduce the effect of the parameters with high sensitivity indices, on the initial transmission. Finally, an optimal control strategy is presented by taking into account the cost associated with control strategies. It is also shown that an optimal control exists for the proposed control problem. The goal of optimal control problem is to minimize, the cost associated with control strategies and the chances of infectious humans, exposed humans and vector population to become infected. Numerical simulations are carried out with the help of Runge-Kutta fourth order procedure.

Sensitivity Analysis and Optimal Control of Anthroponotic Cutaneous Leishmania

This paper is focused on the transmission dynamics and optimal control of Anthroponotic Cutaneous Leishmania. The threshold condition R0 for initial transmission of infection is obtained by next generation method. Biological sense of the threshold condition is investigated and discussed in detail. The sensitivity analysis of the reproduction number is presented and the most sensitive parameters are high lighted. On the basis of sensitivity analysis, some control strategies are introduced in the model. These strategies positively reduce the effect of the parameters with high sensitivity indices, on the initial transmission. Finally, an optimal control strategy is presented by taking into account the cost associated with control strategies. It is also shown that an optimal control exists for the proposed control problem. The goal of optimal control problem is to minimize, the cost associated with control strategies and the chances of infectious humans, exposed humans and vector population to become infected. Numerical simulations are carried out with the help of Runge-Kutta fourth order procedure.

Neural control hierarchy of the heart has not evolved to deal with myocardial ischemia

The consequences of myocardial ischemia are examined from the standpoint of the neural control system of the heart, a hierarchy of three neuronal centers residing in central command, intrathoracic ganglia, and intrinsic cardiac ganglia. The basis of the investigation is the premise that while this hierarchical control system has evolved to deal with "normal" physiological circumstances, its response in the event of myocardial ischemia is unpredictable because the singular circumstances of this event are as yet not part of its evolutionary repertoire. The results indicate that the harmonious relationship between the three levels of control breaks down, because of a conflict between the priorities that they have evolved to deal with. Essentially, while the main priority in central command is blood demand, the priority at the intrathoracic and cardiac levels is heart rate. As a result of this breakdown, heart rate becomes less predictable and therefore less reliable as a diagnostic guide as to the traumatic state of the heart, which it is commonly used as such following an ischemic event. On the basis of these results it is proposed that under the singular conditions of myocardial ischemia a determination of neural control indexes in addition to cardiovascular indexes has the potential of enhancing clinical outcome.

Dynamic neural networking as a basis for plasticity in the control of heart rate

A model is proposed in which the relationship between individual neurons within a neural network is dynamically changing to the effect of providing a measure of "plasticity" in the control of heart rate. The neural network on which the model is based consists of three populations of neurons residing in the central nervous system, the intrathoracic extracardiac nervous system, and the intrinsic cardiac nervous system. This hierarchy of neural centers is used to challenge the classical view that the control of heart rate, a key clinical index, resides entirely in central neuronal command (spinal cord, medulla oblongata, and higher centers). Our results indicate that dynamic networking allows for the possibility of an interplay among the three populations of neurons to the effect of altering the order of control of heart rate among them. This interplay among the three levels of control allows for different neural pathways for the control of heart rate to emerge under different blood flow demands or disease conditions and, as such, it has significant clinical implications because current understanding and treatment of heart rate anomalies are based largely on a single level of control and on neurons acting in unison as a single entity rather than individually within a (plastically) interconnected network.

Cardiac mechanoreceptor function implicated during premature ventricular contraction

In a premature ventricular contraction (PVC), a systolic blood pressure peak is missing during the affected cardiac cycle, leading to a prolonged reduction in blood pressure which is then followed by a large burst of sympathetic outflow. In a normal ventricular contraction, it is generally believed that peak carotid and aortic distensions associated with systolic pressure is the neural feedback that terminates sympathetic outflow through a baroreflex mechanism. Yet, the characteristically large sympathetic burst following a PVC is terminated without a systolic pressure and evidently without this mechanism. To address this anomaly, we examined the possible role of cardiac receptors in providing an alternative mechanism for the termination of sympathetic outflow in a PVC. For this purpose, recordings of electrocardiogram (ECG), arterial blood pressure (ABP), and muscle sympathetic neural activity (MSNA) were made in a human subject during repeated PVC episodes. The time intervals, or "latencies", from key events within the PVC to the peak of the associated MSNA burst were calculated and compared with the latency in a normal ventricular contraction which is associated with central baroreceptor function. It was found that the only event in a PVC that corresponds with a physiologically plausible latency is that which marks the end of ventricular filling. We conclude with the hypothesis that in the unique circumstances of a PVC, where the systolic pressure peak required to trigger arterial baroreceptors to terminate sympathetic outflow is absent, mechanoreceptors in the heart appear to "step in" to perform this sympathoinhibitory function.

Coupled radial and longitudinal displacements and stresses within the arterial wall in pulsatile flow under tethered and free-wall conditions

An analytical solution is presented of the governing equations for the coupled radial and longitudinal displacements and stresses within the finite thickness of the arterial wall in pulsatile flow. The results are used to examine the extent of coupling between the radial and longitudinal dynamics within the vessel wall, particularly when the wall is fully tethered. In the case of a free wall, it is found that the dynamics in the two directions are fairly decoupled from each other when the wavelength is at least of the order of 100 times the vessel radius. At 10 times the vessel radius, however, there is strong coupling between the two. These findings are consistent with expectations in the case of a free wall where the long-wave approximation has been applied in the past. In the case of a tethered wall, however, the results indicate that in general the long-wave approximation is strictly valid only when the combination of wall material and tethering allow the wave to be long.

Neural control of heart rate: the role of neuronal networking

Neural control of heart rate, particularly its sympathetic component, is generally thought to reside primarily in the central nervous system, though accumulating evidence suggests that intrathoracic extracardiac and intrinsic cardiac ganglia are also involved. We propose an integrated model in which the control of heart rate is achieved via three neuronal "levels" representing three control centers instead of the conventional one. Most importantly, in this model control is effected through networking between neuronal populations within and among these layers. The results obtained indicate that networking serves to process demands for systemic blood flow before transducing them to cardiac motor neurons. This provides the heart with a measure of protection against the possibility of "overdrive" implied by the currently held centrally driven system. The results also show that localized networking instabilities can lead to sporadic low frequency oscillations that have the characteristics of the well-known Mayer waves. The sporadic nature of Mayer waves has been unexplained so far and is of particular interest in clinical diagnosis.

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

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

Myogenic activity in autoregulation during low frequency oscillations

Lower body negative pressure (LBNP) was applied in eight human subjects to trigger low frequency oscillations in order to study the nature of functional coupling between the hemodynamic and autonomic nervous systems, with particular focus on how the myogenic response fits within this coupling. To this end muscle sympathetic nerve activity (MSNA), mean arterial pressure (MAP), heart rate (HR), cardiac output (CO), and total peripheral resistance (TPR) were measured at baseline and during LBNP and were then examined in both the time and frequency domains. At the height of low frequency oscillations (~0.1Hz) there was a strong coupling between all the five indices, marked by perfect alignment of their oscillatory frequencies. Results in the time domain show that a fall in MAP is followed by a fall in TPR at 1.58s SD 0.69), a rise in heart rate at 2.64s (SD 0.98), a rise in cardiac output at 3.72s (SD 0.60), a peak in MSNA at 5.71s (SD 1.27) and, finally, a rise in TPR at 7.13s (SD 1.02). A possible interpretation of the latter is that a drop in MAP first triggers a drop in TPR via a myogenic response before the expected rise in TPR via a rise in MSNA. In other words, following a drop in arterial pressure, myogenic response controls vessel diameter before this control is taken over by MSNA. These findings provide a possible resolution of a longstanding conceptual argument against attributing a significant role for the myogenic response in blood flow autoregulation.

Arterial wall tethering as a distant boundary condition

A standing difficulty in the problem of blood vessel tethering has been that only one of the two required boundary conditions can be fully specified, namely, that at the inner (endothelial) wall surface. The other, at the outer layer of the vessel wall, is not known except in the limiting case where the wall is fully tethered such that its outer layer is prevented from any displacement. In all other cases, where the wall is either free or partially tethered, a direct boundary condition is not available. We present a method of determining this missing boundary condition by considering the limiting case of a semi-infinite wall. The result makes it possible to define the degree of tethering imposed by surrounding tissue more accurately in terms of the displacement of the outer layer of the vessel wall, rather than in terms of equivalent added mass which has been done in the past. This new approach makes it possible for the first time to describe the effect of partial tethering in its full range, from zero to full tethering. The results indicate that high tethering leads to high stresses and low displacements within the vessel wall, while low tethering leads to low stresses and high displacements. Since both extremes would be damaging to wall tissue, particularly elastin, this suggest that moderate tethering would be optimum in the physiological setting.

Solutions of the Maxwell viscoelastic equations for displacement and stress distributions within the arterial wall

Mechanical events within the thickness of the vessel wall caused by pulsatile blood flow are considered, with focus on axial dynamics of the wall, driven by the oscillatory drag force exerted by the fluid on the endothelial layer of the wall. It is shown that the focus on the axial direction makes it possible to derive simplified equations of motion which, combined with a viscoelastic model of the wall material, makes it possible in turn to obtain solutions in closed form for the displacement and stress of material elements within the wall. The viscoelastic model allows a study of the dynamics of the wall with different ratios of viscosity to elasticity of the wall material, to mimic changes in the properties of the arterial wall caused by disease or aging. It is found that when the wall is highly viscous the displacements and stresses caused by the flow are confined to a thin layer close to the inner boundary of the wall, while as the wall material becomes less viscous and more rigid the displacements and stresses spread deeper into the thickness of the wall to affect most of its elements.

Role of vascular bed compliance in vasomotor control in human skeletal muscle

The current view of neurogenic vasomotor control in skeletal muscle is based largely on changes in vascular bed resistance. The purpose of this study was to determine to what extent vascular bed compliance may also play a role in this regulation. For this purpose, pressure waveforms (Millar and Finometer) and flow waveforms (Doppler ultrasound) were measured simultaneously in the brachial artery of seven healthy individuals during physiological manoeuvres which were expected to produce non-neurogenic changes in resistance (wrist-cuff occlusion; n = 5) or compliance (arm elevation; n = 6) of the forearm vascular bed. Vascular resistance (R) was calculated from the average flow and pressure values. A lumped Windkessel model was used to obtain vascular bed compliance (C) from these concurrently measured waveforms. Compared with baseline (3.81 +/- 1.59 ml min(-1) mmHg(-1)), wrist occlusion increased R (65 +/- 75%; P < 0.05) with minimal change in C (-15 +/- 16%; n.s.). Compared with the arm in neutral position (0.0075 +/- 0.003 ml mmHg(-1)), elevation of the arm above heart level produced a 86 +/- 41% increase in C (P < 0.05) with little change in R (-5 +/- 11%). In addition, neurogenic changes were assessed during lower body negative pressure (LBNP) and a cold pressor test (CPT; n = 7). Lower body negative pressure induced a 29 +/- 24% increase in R and a 26 +/- 12% decrease in C (both P < 0.05). The CPT induced no consistent change in R but a 22 +/- 7% reduction in C (P < 0.05). It was concluded that vascular bed compliance is an independent variable which should be considered along with vascular bed resistance in the mechanics of vasomotor regulation in skeletal muscle.

Mechanism of smart baroreception in the aortic arch

A mechanism is proposed by which the patch of baroreceptors along the inner curvature of the arch of the aorta can sense hemodynamic events occurring downstream from the aortic arch, in the periphery of the arterial tree. Based on a solution of equations governing the elastic movements of the aortic wall, it is shown that the pressure distribution along the patch of baroreceptors has the same functional form as the distribution of strain along the patch. The significance of these findings are discussed, particularly as they relate to the possibility of a neuromechanical basis of essential hypertension.

"Smart" baroreception along the aortic arch, with reference to essential hypertension

Beat-to-beat regulation of heart rate is dependent upon sensing of local stretching or local "disortion" by aortic baroreceptors. Distortions of the aortic wall are due mainly to left ventricular output and to reflected waves arising from the arterial tree. Distortions are generally believed to be useful in cardiac control since stretch receptors or aortic baroreceptors embedded in the adventitia of the aortic wall, transduce the distortions to cardiovascular neural reflex pathways responsible for beat-to-beat regulation of heart rate. Aortic neuroanatomy studies have also found a continuous strip of mechanosensory neurites spread along the aortic inner arch. Although their purpose is now unknown, such a combined sensing capacity would allow measurement of the space and time dependence of inner arch wall distortions due, among other things, to traveling waves associated with pulsatile flow in an elastic tube. We call this sensing capability--"smart baroreception." In this paper we use an arterial tree model to show that the cumulative effects of wave reflections, from many sites far downstream, have a surprisingly pronounced effect on the pressure distribution in the root segment of the tree. By this mechanism global hemodynamics can be focused by wave reflections back to the aortic arch, where they can rapidly impact cardiac control via smart baroreception. Such sensing is likely important to maintain efficient heart function. However, alterations in the arterial tree due to aging and other natural processes can lead in such a system to altered cardiac control and essential hypertension.

Effects of stent stiffness on local haemodynamics with particular reference to wave reflections

The placement of a rigid stent within an elastic vessel produces wave reflection sites at the entrance to and exit from the stent. The net haemodynamic effects of these reflections depend critically on the degree of stiffness of the stent and on its length and position within the diseased vessel, variables that have been found to affect the clinical performance of a stent. Here these effects are examined analytically, using a segmented tube model. The results indicate that the presence of the stent within the larger diseased vessel has the effect of producing higher pressure at the vessel entrance than that at exit. This pressure difference, when superimposed on the underlying pressure distribution within the vessel, has the net effect of actually aiding rather than impeding the flow, but the extent of this depends on the length and position of the stent. A short stent placed near the entrance of the diseased vessel may be favoured clinically for producing the least perturbation in the underlying haemodynamics and thus reducing the chance of restenosis, while a long stent placed near the exit may be favoured for producing a positive pressure difference and thus aiding the flow.

Smaller, stiffer coronary bypass can moderate or reverse the adverse effects of wave reflections

The question of whether the mechanical stiffness of a coronary bypass or that of a diseased coronary artery can have a significant effect on the hemodynamics in these vessels is addressed analytically, with emphasis on the effects of wave reflections. The analysis is based on a model of the vessels involved, and the results show the essential hemodynamic effects in each vessel. It is found that in the absence of a bypass graft, wave reflections resulting from a narrowing and stiffening of a diseased coronary artery have the effect of actually aiding the flow in the diseased vessel. In the presence of a bypass graft, however, the effects of wave reflections are reversed and become adverse to flow in both the bypass graft and the diseased coronary artery. A stiffer bypass moderates these effects and is therefore preferable to a more elastic bypass. The adverse effects also depend critically on the relative diameter of the bypass. Here the results indicate that a bypass of smaller diameter than that of the native coronary artery can moderate and even reverse the adverse effects of wave reflections resulting from the presence of the bypass.

Fractal dimensions and multifractility in vascular branching

A definition for the fractal dimension of a vascular tree is proposed based on the hemodynamic function of the tree and in terms of two key branching parameters: the asymmetry ratio of arterial bifurcations and the power law exponent governing the relation between vessel diameter and flow. Data from the cardiovascular system, which generally exhibit considerable scatter in the values of these two parameters, are found to produce the same degree of scatter in the value of the fractal dimension. When this scatter is explored for a multifractal pattern, however, it is found that the required collapse onto a single curve is achieved in terms of the coarse Hölder exponent. Thus, the presence of multifractility is confirmed, and the legitimacy of the defined dimension is affirmed in the sense of the theoretical Hausdorff limit in as much as this limit can be reached with experimental data.

Arterial branching within the confines of fractal L-system formalism

Parametric Lindenmayer systems (L-systems) are formulated to generate branching tree structures that can incorporate the physiological laws of arterial branching. By construction, the generated trees are de facto fractal structures, and with appropriate choice of parameters, they can be made to exhibit some of the branching patterns of arterial trees, particularly those with a preponderant value of the asymmetry ratio. The question of whether arterial trees in general have these fractal characteristics is examined by comparison of pattern with vasculature from the cardiovascular system. The results suggest that parametric L-systems can be used to produce fractal tree structures but not with the variability in branching parameters observed in arterial trees. These parameters include the asymmetry ratio, the area ratio, branch diameters, and branching angles. The key issue is that the source of variability in these parameters is not known and, hence, it cannot be accurately reproduced in a model. L-systems with a random choice of parameters can be made to mimic some of the observed variability, but the legitimacy of that choice is not clear.

On fractal properties of arterial trees

The question of fractal properties of arterial trees is considered in light of data from the extensive tree structure of the right coronary artery of a human heart. Because of the highly non-uniform structure of this tree, the study focuses on the purely geometrical rather than statistical aspects of fractal properties. The large number of arterial bifurcations comprising the tree were found to have a mixed degree of asymmetry at all levels of the tree, including the depth of the tree where it has been generally supposed that they would be symmetrical. Cross-sectional area ratios of daughter to parent vessels were also found to be highly mixed at all levels, having values both above and below 1.0, rather than consistently above as has been generally supposed in the past. Calculated values of the power law index which describes the theoretical relation between the diameters of the three vessel segments at an arterial bifurcation were found to range far beyond the two values associated with the cube and square laws, and not clearly favoring one or the other. On the whole the tree structure was found to have what we have termed "pseudo-fractal" properties, in the sense that vessels of different calibers displayed the same branching pattern but with a range of values of the branching parameters. The results suggest that a higher degree of fractal character, one in which the branching parameters are constant throughout the tree structure, is unlikely to be attained in non-uniform vascular structures.

Pulsatile flow in tubes of elliptic cross sections

The compression of blood vessels by surrounding tissue is an important problem in hemodynamics, most prominently in studies relating to the heart. In this study we consider a long tube of elliptic cross section as an idealization of the geometry of a compressed blood vessel. An exact solution of the governing equations for pulsatile flow in a tube of elliptic cross section involves Mathieu functions which are considerably more difficult to evaluate than the Bessel functions in the case of a circular cross section. Results for the velocity field, flow rate and wall shear stress are obtained for different values of the pulsation frequency and ellipticity, with emphasis on how the effects of frequency and ellipticity combine to determine the flow characteristics. It is found that in general the effects of ellipticity are minor when frequency is low but become highly significant as the frequency increases. More specifically, the velocity profile along the major axis of the elliptic cross section develops sharp double peaks; the flow rate is reduced in approximately the same proportion as in the case of circular cross section; and the point of maximum shear on the tube wall migrates away from the minor axis where it is located in steady flow.

Mechanics of blood supply to the heart: wave reflection effects in a right coronary artery

Mechanics of blood flow in the coronary circulation have in the past been based largely on models in which the detailed architecture of the coronary network is not included because of lack of data: properties of individual vessels do not appear individually in the model but are represented collectively by the elements of a single electric circuit. Recent data from the human heart make it possible, for the first time, to examine the dynamics of flow in the coronary network based on detailed, measured vascular architecture. In particular, admittance values along the full course of the right coronary artery are computed based on actual lengths and diameters of the many thousands of branches which make up the distribution system of this vessel. The results indicate that effects of wave reflections on this flow are far more significant than those generally suspected to occur in coronary blood flow and that they are actually the reverse of the well known wave reflection effects in the aorta.

Tree structure and branching characteristics of the right coronary artery in a right-dominant human heart

The hierarchic branching tree structure of the right coronary artery in a right-dominant human heart is examined in terms of detailed measurements of lengths and diameters of the individual vessel segments comprising the structure. The results show that, while the right coronary artery in this heart does not supply a large portion of the posterior left ventricular wall, it nevertheless serves the left side of the heart to a greater extent than it does the right side in terms of the number and volume of vessel segments involved. The results show further that on the right side of this heart the right coronary artery exhibits clear features of a "distributing' vessel, but these features change abruptly as the vessel reaches the small region of the posterior left ventricular wall which it serves.

Pressure peaking in pulsatile flow through arterial tree structures

An analytical iterative scheme is presented for computing the local characteristics of pressure and flow waves as they progress along a tree structure and become modified by wave reflections. Results are obtained to illustrate the phenomenon of pressure peaking under two different sets of circumstances. In the first case, the propagation of a single harmonic wave along a simple tree is considered, where wave reflections modify the amplitude of the pressure wave as it travels. In the second case, the propagation of a composite wave along a tree with multiple branches is considered, where wave reflections modify the shape of the wave as it travels and cause it to peak. The results demonstrate unambiguously that the root cause of this phenomenon is wave reflections caused by stepwise decreases in admittance, as has been previously suggested, rather than due to nonlinear interactions, as has also been previously suggested. It is shown clearly that even when wave reflections combine linearly, they lead to considerable peaking in the pressure waveform.

Mechanics of wave reflections in a coronary bypass loop model: the possibility of partial flow cut-off

Local hemodynamics in a bypass loop model with dimensions typical of those in the human coronary circulation are studied, particularly with regards to the effects of wave reflections. While in a single vessel of such dimensions and a single source of wave reflections the effects would be insignificant, in a bypass loop the combination of a narrowed vessel with a converging junction and several reflection sites may produce wave reflections with significant effects on the pressure and flow distributions within the vessels forming the loop. Calculations to test this possibility were performed, based on D'Alembert's solutions of the wave equation and with a proposed matching scheme to deal with the converging junction. The results indicate that there are very large wave reflection effects at a frequency of 10 Hz, smaller but significant effects at 5 Hz, and some insignificant effects at 1 Hz. The results also indicate that partial flow cut-off may occur within the loop under some singular circumstances, whereby certain harmonic components of the incident wave are totally reflected. In the clinical setting these effects would detract from the efficiency of the bypass as a conduit, to a degree dependent on the degree of occlusion of the bypassed vessel. The choice of a larger diameter for the bypass appears to diminish this dependence and is thus on the whole favorable despite the contribution it makes to impedance mismatch at the junction.

Reflection coefficients in pulsatile flow through converging junctions and the pressure distribution in a simple loop

Analytical expressions for the reflection coefficients in pulsatile flow through converging junctions are derived by two independent methods and are used to study the effects of wave reflections on the pressure distribution in a simple vascular loop. A simulated physiological situation is used as an example in which the loop is formed by the combination of a bypass and a bypassed vessel, the relative diameter of the latter being varied in order to simulate a narrowing. The results demonstrate how, in the case of a converging junction, the effects of wave reflections on the pressure distribution in one vessel depend on conditions within the vessel itself as well as in the other. The new reflection coefficients take into account this interdependence of flow in the two vessels forming a converging junction, and are shown to be consistent with reflection coefficients commonly used in diverging junctions.

Velocity profiles and phasic flow patterns in the non-stenotic human left anterior descending coronary artery during cardiac surgery

OBJECTIVE

The aim was to investigate the phasic characteristics of normal human left coronary artery flow and velocity profiles across the vessel.

METHODS

The phasic characteristics of flow in the human left anterior descending coronary artery, the centreline flow velocities, and the velocity profiles were measured in 10 patients during corrective surgery for atrial septal defect after closure of the defect. None of these patients had any detectable coronary artery stenosis or left ventricular hypertrophy. Measurements were made with a 20 MHz 80 channel pulsed Doppler velocimeter.

RESULTS

The velocity waveform displayed a diastolic-predominant pattern with a systolic to diastolic velocity ratio of 0.29(SD 0.17). Reverse flow was observed in early systole in five patients and in mid to late systole in six patients. The values of peak Reynolds number, unsteadiness parameter, and pulsatility index were 504(198), 2.5(0.6), and 5.9(4.4) respectively. The velocity profiles during diastole showed considerable variability in shape, ranging from symmetrical to skewed to M shaped patterns. The peak wall shear rate was 765(250) s-1 on the epicardial wall of the vessel and 712(301) s-1 on the myocardial wall; the difference was not statistically significant.

CONCLUSIONS

The velocity waveform displayed a diastolic-predominant pattern. Considerable variability in shape of the velocity profile was found and was perhaps due to the time evolution of the velocity profile within the diastolic time period.

[Vascular radiology and physics of fluids]

In the cardiovascular system, morphology and functionality are closely related. The observation that atherosclerotic lesions appear with different incidence in the various sites and prefer bifurcations, suggested that vascular architecture might be an important pathogenic factor. Vascular geometry deals with the evaluation of the "form" element from a theoretical point of a view, pointing out the angles and diameters which can make the arterial system physiologically more efficient. The key element of the system is the bifurcation: depending on whether bifurcations are considered as a single entity or as a whole, either "local" or "global" geometry is employed. The morphology of the bifurcation directly affects blood flow patterns, influencing both blood flow velocity and wall shear stress. Experimental studies evidenced that high blood flow velocities and low wall shear stress have, respectively, a preventive or favorable role in atheroma development. By observing these altered flow sites, the areas which are most likely to develop atherosclerotic damage have been detected in the various bifurcations; the areas correspond to angiographic and autoptic findings. Based on their experience, the authors discuss the contribution that the different imaging techniques can give to the study of vascular geometry.

Relation between diameter and flow in major branches of the arch of the aorta

In the analysis of arterial branching the classical "cube law' has provided a working model for the relation between the diameter of a blood vessel and the flow which the vessel carries on a long-term basis. The law has shown good agreement with biological data, but questions remain regarding its applicability to all levels of the arterial tree. The present study tests the hypothesis that the cube law may not be valid in the first few generations of the arterial tree, where vessel capacitance and gross anatomy may play important roles. Biological data have shown some support for this hypothesis in the past but the heterogeneity characteristic of past data has not allowed a conclusive test so far. We present new data which have been obtained from the same location on the arterial tree and in sufficient number to make this test possible for the first time. Also, while past tests have been based primarily on correlation of the measured data with an assumed power law, we show here that this can be misleading. The present data allow a simpler test which does not involve correlation and which leads to more direct conclusions. For the vessels surveyed, the results show unequivocally that the relation between diameter and flow is governed by a 'square law' rather than the classical cube law. Coupled with past findings this suggests that the square law may apply at the first few levels of the arterial tree, while the cube law continues from there to perhaps the precapillary levels.

'Hemorrhagic' and microvascular phenomena within the arterial wall

OBJECTIVE

To study microvasculature and hemorrhage within the arterial wall.

DESIGN

Human autopsy specimens of the arch of the aorta, the carotid, brachiocephalic, subclavian and coronary arteries perfused with liquid casting material and maintained at physiological pressure until the material had set.

MAIN RESULTS

Evidence of dense microvasculature and other phenomena which have the morphological appearance of 'hemorrhage' within the arterial wall, coupled with calcified matter and other impressions made on the cast by atherosclerotic plaques.

CONCLUSIONS

The findings lend support to suggestions in the literature that neovascularization may play a role in the pathogenesis of coronary atherosclerosis and its sequelae, that hemorrhage into the intima may be due to rupture of capillaries which are derived from the coronary lumen, and that an increase in microvasculature occurs in the immediate vicinity of localized atherosclerotic lesions.

Origin of the brachiocephalic trunk, left carotid, and left subclavian arteries from the arch of the human aorta

We measured branching angles of the brachiocephalic trunk, left carotid, and left subclavian arteries at their points of origin from the arch in 117 human specimens. We compared the measured angles with theoretical values commonly used in the study of arterial branching, and examined possible correlations with age and with the presence of atherosclerotic lesions. The results may be useful in the modelling and analysis of hemodynamic factors in the pathogenesis of vascular disease at these junctions and may be of some interest in clinical angiography. The junctions are important gateways for blood supply to the head and upper limbs and often are involved with lesions that affect blood supply to the brain.

Continuum analysis of common branching patterns in the human arch of the aorta

A model is proposed for describing common variations in the arrangement of branches on the arch of the human aorta, and the model is used to analyze data from 123 human arches. The analysis allows the observed variations to fall freely along a continuous spectrum, rather than be confined to discrete categories as is commonly done at present. The results thus describe these variations in a more natural way and throw some new light on their likely source.

Roots and calibers of the human coronary arteries

The anatomical structure of the coronary-aortic junctions in humans is studied by using corrosion casts of the coronary network. A model is proposed for the specification of these junctions in terms of vessel diameters and branching angles, and the model is used to produce morphological data on these junctions which hitherto have not been available. This anatomical model correlates poorly with the accepted theoretical model of arterial bifurcations in the cardiovascular system. The results suggest that the structure of the coronary-aortic junctions is very different from the structure of typical arterial bifurcations and, by implication, that the flow conditions under which they function are very different. A good understanding of these junctions is important in coronary bypass surgery, where the coronary-aortic junctions are emulated by creating a new anastomosis for the graft at the base of the ascending aorta, and in coronary artery disease, where atherosclerotic lesions occur not far from the coronary-aortic junctions.

Distributing and delivering vessels of the human heart

The branching characteristics of the right coronary artery, acute marginal, posterior descending, left anterior descending, circumflex, and obtuse marginal arteries are compared with those of diagonal branches, left and right ventricular branches, septal, and higher-order branches, to test a newly proposed functional classification of the coronary arteries in which the first group rank as distributing vessels and the second as delivering vessels. According to this classification, the function of the first type is merely to convey blood to the borders of myocardial zones, while the function of the second is to implement the actual delivery of blood into these zones. This functional difference is important in the hemodynamic analysis of coronary heart disease, as it provides an assessment of the role of a vessel within the coronary network and hence an assessment of the functional importance of that vessel in a particular heart. Measurements from casts of human coronary arteries are used to examine the relevant characteristics of these vessels and hence to test the basis of this classification.

Network analysis of an arterial tree

The arterial tree of a Sprague-Dawley rat was casted and carefully mapped with the aim of comparing its network characteristics with those suggested by the classical model of an arterial tree. It is shown that if the tree is to be measured accurately, the concept of 'whole vessels' on which the classical model is based must be abandoned since such vessels do not actually exist in the network, nor can they be accurately defined. The concept of 'vessel segments' is proposed instead and its use is demonstrated. A total of 1313 vessel segments in the arterial tree of the rat are mapped and divided into well defined 'levels'. The length and diameter of each segment are measured and the distribution and averages of these at different levels are presented as indicators of the branching characteristics of the tree.

Morphometric analysis of the distributing vessels of the kidney

Branching angles and branch diameters of the distributing vessels in the renal networks of rats were measured and the results are compared with data reported previously from the coronary network of the same species. Comparison is also made with what is known to be optimum on theoretical grounds to determine to what extent the branching characteristics of the renal network are governed by considerations of optimality, and to what extent they are affected by other considerations, relating particularly to the role that the network plays in the blood processing function of the kidney.

Segment analysis of human coronary arteries

Detailed measurements of vessel lengths and diameters from the coronary network of the human heart are presented. To allow accurate definition of length and diameter, the measurements were made in terms of vessel segments rather than whole vessels. A vessel segment was defined to be the interval between two consecutive branching sites. The results provide the first quantitative description of the coronary network. The new concept of vessel segment and the method of analysis are proposed as a means for an accurate description of the branching characteristics of the coronary network, comparing the network with others in the cardiovascular system and comparing the vasculature in the normal versus that in the diseased heart.

Branching characteristics of human coronary arteries

Branching angles and branch diameters were measured in a total of 850 arterial junctions in the coronary networks of two human hearts. Comparison is made with similar data obtained previously from the coronary networks of rats, and with what is considered to be optimum on theoretical grounds. It is concluded that the branching characteristics of the human coronary arteries are closer to the theoretical optimum than those of the coronary networks of rats. While the human data exhibit some departure from optimality and a good amount of scatter, these are well within levels observed elsewhere in the cardiovascular systems of man and animals, and considerably better than those found in the coronary networks of rats. The departure from optimality, in terms of physiological cost to the system, is within 5% for most data points.

Cost analysis of arterial branching in the cardiovascular systems of man and animals

A new scheme is presented whereby data on arterial branching can be interpreted in terms of direct cost to the physiological system. The scheme makes it possible to assess, at a glance, the true degree of optimality of an arterial network. Departure from optimality is indicated in terms of cost, rather than in terms of the difference between theoretical and measured branching angles. The scheme is applied to several groups of biological data and new conclusions are reached with regard to their degrees of optimality.

Morpho-functional anatomy of the human coronary arteries with reference to myocardial ischemia

The vascular beds of 42 human hearts were examined to determine the underlying morpho-functional arrangement of the coronary arteries. The hearts were obtained at autopsy from cases where the immediate cause of death was not related to cardiovascular disease. It was found that while the course and size of individual vessels vary considerably from heart to heart, there is an underlying pattern in the functional arrangement of the vessels. On the basis of this pattern, a new functional classification of coronary arteries is proposed, together with a related concept of intrinsic "zones" of the myocardium. It is concluded that a deficiency in coronary blood supply in ischemic heart disease would be more accurately described in terms of the affected zones of the myocardium rather than the affected vessels. It is the effect of stenosis, not the stenosis per se, which must be assessed and documented ultimately. A chart is proposed to facilitate that documentation.