M. Zamir

Optimality principles in arterial branching

Abstract A comparative study of four optimality principles for the branching geometry of blood arteries is presented. The results offer four different criteria which can be tested by experimental data to establish which of these principles is followed in the cardiovascular system. More significantly, the results suggest the further possibility that the geometry of arterial junctions may be governed by all of these principles simultaneously, to thus achieve a much higher degree of optimality than has hitherto been suspected. This result offers a basis for seeking a correlation between the degree of optimality of a particular junction and the incidence of certain arterial lesions at that junction.

Cost of departure from optimality in arterial branching

Measurements of branching angles in the arterial tree have in the past indicated a great deal of scatter away from what is expected to be optimum on theoretical grounds. In this study the cost penalty of nonoptimum branching angles is calculated for the first time to determine how far from optimum these angles are. The results lead to the remarkable conclusion that while the scatter of the measured branching angles is fairly large, they represent deviations from optimum angles which correspond to only 2% or so penalty in cost.

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.

Pulsatile Flow in Tubes of Elliptic Cross Sections

AbstractThe 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.

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.

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.

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.

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 phenoenon 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.

Effects of infection on honey bee population dynamics: a model.

We propose a model that combines the dynamics of the spread of disease within a bee colony with the underlying demographic dynamics of the colony to determine the ultimate fate of the colony under different scenarios. The model suggests that key factors in the survival or collapse of a honey bee colony in the face of an infection are the rate of transmission of the infection and the disease-induced death rate. An increase in the disease-induced death rate, which can be thought of as an increase in the severity of the disease, may actually help the colony overcome the disease and survive through winter. By contrast, an increase in the transmission rate, which means that bees are being infected at an earlier age, has a drastic deleterious effect. Another important finding relates to the timing of infection in relation to the onset of winter, indicating that in a time interval of approximately 20 days before the onset of winter the colony is most affected by the onset of infection. The results suggest further that the age of recruitment of hive bees to foraging duties is a good early marker for the survival or collapse of a honey bee colony in the face of infection, which is consistent with experimental evidence but the model provides insight into the underlying mechanisms. The most important result of the study is a clear distinction between an exposure of the honey bee colony to an environmental hazard such as pesticides or insecticides, or an exposure to an infectious disease. The results indicate unequivocally that in the scenarios that we have examined, and perhaps more generally, an infectious disease is far more hazardous to the survival of a bee colony than an environmental hazard that causes an equal death rate in foraging bees.

Vagal Nerve Stimulation Therapy: What Is Being Stimulated?

Vagal nerve stimulation in cardiac therapy involves delivering electrical current to the vagal sympathetic complex in patients experiencing heart failure. The therapy has shown promise but the mechanisms by which any benefit accrues is not understood. In this paper we model the response to increased levels of stimulation of individual components of the vagal sympathetic complex as a differential activation of each component in the control of heart rate. The model provides insight beyond what is available in the animal experiment in as much as allowing the simultaneous assessment of neuronal activity throughout the cardiac neural axis. The results indicate that there is sensitivity of the neural network to low level subthreshold stimulation. This leads us to propose that the chronic effects of vagal nerve stimulation therapy lie within the indirect pathways that target intrinsic cardiac local circuit neurons because they have the capacity for plasticity.