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.     

Optimality principles in arterial branching

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.     

The cost of departure from optimal radii in microvascular networks

In the Murray optimality model of branching vasculatures, the radii of vessels are related to blood viscosity, vascular metabolic rate, and blood flow rate, in such a way as to minimize the total work (hydraulic and metabolic) of the system. The model predicts that flow is proportional to the cube of a vessel radius, and that at junctions the cube of the radius of the parent vessel equals the sum of the cubes of the daughter radii. In comparing real vasculatures to the Murray model, we have previously had no expressions for evaluating the apparent energy cost for departures from the optimal junction exponent of 3. Such expressions are derived here. They show that junction exponents, from about 1.5 to large positive values, are within 5% of the energy minimum. With the new equations, observed individual junctions or entire vascular trees can be compared, energy-wise, with the Murray optimum. Junctions in the transverse arteriolar trees of cat sartorius muscle were compared to the Murray optimality model, using these new expressions. The junction exponents for these small pre-capillary vessels had a broad range, with a median value greater than the Murray optimum of 3. The exponents were restricted, however, to values requiring, at individual junctions, little increase in energy. The majority of junctions had energy costs less than 1% above the Murray minimum. For entire trees involving many junctions the departures from optimality averaged less than 10%. Thus, while the branching geometry for these microvascular trees deviates significantly from the Murray optimum in the direction of larger daughter to parent ratios, the departures are small in energy terms.     

Nonsymmetrical bifurcations in arterial branching

The results of optimality studies of the branching angles of arterial bifurcations are extended to nonsymmetrical bifurcations. Predicted nonsymmetrical bifurcations are found to be not unlike those observed in the cardiovascular system.     

Finding the optimal lengths for three branches at a junction

This paper presents an exact analytical solution to the problem of locating the junction point between three branches so that the sum of the total costs of the branches is minimized. When the cost per unit length of each branch is known the angles between each pair of branches can be deduced following reasoning first introduced to biology by Murray. Assuming the outer ends of each branch are fixed, the location of the junction and the length of each branch are then deduced using plane geometry and trigonometry. The model has applications in determining the optimal cost of a branch or branches at a junction. Comparing the optimal to the actual cost of a junction is a new way to compare cost models for goodness of fit to actual junction geometry. It is an unambiguous measure and is superior to comparing observed and optimal angles between each daughter and the parent branch. We present data for 199 junctions in the pulmonary arteries of two human lungs. For the branches at each junction we calculated the best fitting value of x from the relationship that flow ∞ (radius)^x. We found that the value of x determined whether a junction was best fitted by a surface, volume, drag or power minimization model. While economy of explanation casts doubt that four models operate simultaneously, we found that optimality may still operate, since the angle to the major daughter is less than the angle to the minor daughter. Perhaps optimality combined with a space filling branching pattern governs the branching geometry of the pulmonary artery.     

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.     

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.     

Branching pattern of pulmonary arterial tree in anesthetized dogs

The geometry of the pulmonary arterial tree of six adult dogs was measured by a high-speed, volume-scanning, X-ray tomographic technique. After the dogs were anesthetized a catheter was advanced to the right ventricular outflow tract and 2 mL/kg Renovist contrast agent injected rapidly. During the subsequent pulmonary arterial phase of the angiogram the dogs were scanned. Three-dimensional geometry of the pulmonary arterial tree was measured in terms of vessel segment cross-sectional area, branching angles and interbranch segment lengths along axial pathways. The effect of lung inflation and phase of the cardiac cycle on geometry was shown to be most marked on vessel cross-sectional area. The geometric branching patterns in all dogs were similar. The observed, in-vivo branching pattern behaved somewhat like the branching pattern predicted from optimized models proposed by Murray, Zamir, and Uylings.     

The role of shear forces in arterial branching

A new optimality principle for the branching angles of blood vessels in the cardiovascular system is proposed: the principle of minimum drag. The results are examined in the light of general observations and compared with those obtained from the principles of minimum work and minimum volume. It is shown that in some aspects the new principle is equally consistent with observations, and, in other aspects, it is perhaps more plausible than the other two principles.     

Sequential Analysis of Longitudinal Data in a Prospective Nested Case–Control Study

Summary The nested case–control design is a relatively new type of observational study whereby a case–control approach is employed within an established cohort. In this design, we observe cases and controls longitudinally by sampling all cases whenever they occur but controls at certain time points. Controls can be obtained at time points randomly scheduled or prefixed for operational convenience. This design with longitudinal observations is efficient in terms of cost and duration, especially when the disease is rare and the assessment of exposure levels is difficult. In our design, we propose sequential sampling methods and study both (group) sequential testing and estimation methods so that the study can be stopped as soon as the stopping rule is satisfied. To make such a longitudinal sampling more efficient in terms of both numbers of subjects and replications, we propose applying sequential sampling methods to subjects and replications, simultaneously, until the information criterion is fulfilled. This simultaneous sequential sampling on subjects and replicates is more flexible for practitioners designing their sampling schemes, and is different from the classical approaches used in longitudinal studies. We newly define the σ‐field to accommodate our proposed sampling scheme, which contains mixtures of independent and correlated observations, and prove the asymptotic optimality of sequential estimation based on the martingale theories. We also prove that the independent increment structure is retained so that the group sequential method is applicable. Finally, we present results by employing sequential estimation and group sequential testing on both simulated data and real data on children's diarrhea.     

Adaptive regulation of wall shear stress optimizing vascular tree function

The branching structure of the mammalian arterial tree has been known to be close to that of an optimal conduit system of the minimum work model characterized as the branch system of constant wall shear rate. The physiological mechanism producing such construction was considered to be based on the local response of arterial caliber induced by the wall shear stress (shear rate × blood viscosity) and thereby maintaining this stress constant, which was previously observed at the canine common carotid artery shunted to the external jugular vein. The stress levels at various parts of the arterial system estimated from available data fell within ±50% of the mean (15 dyn/cm²), which was consistent with the value predicted from the model. Theoretical analyses on the cost function of the model indicated that the suspected variation of shear rate levels in the arterial tree due to the anomalous changes in blood viscosity which might bring about 3- to 4-fold differences between the minimum and maximum shear rates would cause less than 10% increase in the total energy cost. It was concluded that a local adaptive response to wall shear stress is the mechanism which effectively optimizes the design of the arterial tree.     

Local geometry of arterial branching

A model of the geometrical structure of arterial bifurcations is proposed in the context of optimality of the bifurcation as a fluid conducting system. Optimality is considered both globally, in terms of the cardiovascular system as a whole, and locally, in terms of the orderliness of the flow in the bifurcation region. It is shown that a bifurcation can be optimal both globally and locally. Typical examples of such bifurcations are given.     

Modeling of branching structures of plants

Previous studies of branching structures generally focused on arteries. Four cost models minimizing total surface area, total volume, total drag and total power losses at a junction point have been proposed to study branching structures. In this paper, we highlight the branching structures of plants and examine which model fits data of branching structures of plants the best. Though the effect of light (e.g. phototropism) and other possible factors are not included in these cost models, a simple cost model with physiological significance, needs to be verified before further research on modeling of branching structures is conducted. Therefore, data are analysed in this paper to determine the best cost model. Branching structures of plants are studied by measuring branching angles and diameters of 234 junctions from four species of plants. The sample includes small junctions, large junctions, two- and three-dimensional junctions, junctions with three branches joining at a point and those with four branches joining at a point. First, junction exponents (x) were determined. Second, log-log plots indicate that model of volume minimization fits data better than other models. Third, one-sided t -tests were used to compare the fitness of four models. It is found that model of volume minimization fits data better than other cost models.     

Learning the optimal control of coordinated eye and head movements

Various optimality principles have been proposed to explain the characteristics of coordinated eye and head movements during visual orienting behavior. At the same time, researchers have suggested several neural models to underly the generation of saccades, but these do not include online learning as a mechanism of optimization. Here, we suggest an open-loop neural controller with a local adaptation mechanism that minimizes a proposed cost function. Simulations show that the characteristics of coordinated eye and head movements generated by this model match the experimental data in many aspects, including the relationship between amplitude, duration and peak velocity in head-restrained and the relative contribution of eye and head to the total gaze shift in head-free conditions. Our model is a first step towards bringing together an optimality principle and an incremental local learning mechanism into a unified control scheme for coordinated eye and head movements.     

A new fundamental bioheat equation for muscle tissue--part II: Temperature of SAV vessels

In this study, a new theoretical framework was developed to investigate temperature variations along countercurrent SAV blood vessels from 300 to 1000 microm diameter in skeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaum-Jiji bioheat equation and, heretofore, have been treated using discrete numerical methods. A new tissue cylinder surrounding these vessel pairs is defined based on vascular anatomy, Murray's law, and the assumption of uniform perfusion. The thermal interaction between the blood vessel pair and surrounding tissue is investigated for two vascular branching patterns, pure branching and pure perfusion. It is shown that temperature variations along these large vessel pairs strongly depend on the branching pattern and the local blood perfusion rate. The arterial supply temperature in different vessel generations was evaluated to estimate the arterial inlet temperature in the modified perfusion source term for the s vessels in Part I of this study. In addition, results from the current research enable one to explore the relative contribution of the SAV vessels and the s vessels to the overall thermal equilibration between blood and tissue.     

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.     

Dominance of geometric overelastic factors in pulse transmission through arterial branching

The branching characteristic of the arterial system is such that blood pressure pulses propagate with minimum loss. This characteristic depends on the geometric and elastic properties of branching vessels. In the current investigation, mathematical relations of branching geometry and elastic properties are formulated and their relative contributions to pulse reflection at an arterial junction are analyzed. Results show that alteration of pulse transmission through the junction is more significantly affected by changes in branching vessel radii and wall thickness than by corresponding percentage changes in vessel wall elastic moduli.     

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.     

A model for renal arterial branching based on graph theory

The kidney is one of the most complicated organs in terms of structure and physiology, in part because it is highly vascularized. The renal vascular development occurs through two mechanisms that sometimes overlap: vasculogenesis and angiogenesis. Here, we consider angiogenesis to model the renal arterial tree with the two processes of vascular angiogenesis: sprouting and splitting. We recognize the vessels are not tubes with ends that get glued but physiological factors are relevant into the vascular development. Our contribution integrates the graph theory and physiological information to derive a quantitative model for the vascular tree in the sense that the vertices and edges represent, respectively, a branching point and a vessel. From such a premise, development of the arterial vascular tree of the kidney is mathematically expressed, including physiological processes as the effect of the vascular endothelial growth factor (VEGF) on the vessel length. A definition of the graph is used to visualize the topology of vascular tree in kidney providing physiological information into the edges. Thus, renal arterial branching is modeled as a graph where edges are labeled and oriented.     

The inactivation principle: mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements

An important question in the literature focusing on motor control is to determine which laws drive biological limb movements. This question has prompted numerous investigations analyzing arm movements in both humans and monkeys. Many theories assume that among all possible movements the one actually performed satisfies an optimality criterion. In the framework of optimal control theory, a first approach is to choose a cost function and test whether the proposed model fits with experimental data. A second approach (generally considered as the more difficult) is to infer the cost function from behavioral data. The cost proposed here includes a term called the absolute work of forces, reflecting the mechanical energy expenditure. Contrary to most investigations studying optimality principles of arm movements, this model has the particularity of using a cost function that is not smooth. First, a mathematical theory related to both direct and inverse optimal control approaches is presented. The first theoretical result is the Inactivation Principle, according to which minimizing a term similar to the absolute work implies simultaneous inactivation of agonistic and antagonistic muscles acting on a single joint, near the time of peak velocity. The second theoretical result is that, conversely, the presence of non-smoothness in the cost function is a necessary condition for the existence of such inactivation. Second, during an experimental study, participants were asked to perform fast vertical arm movements with one, two, and three degrees of freedom. Observed trajectories, velocity profiles, and final postures were accurately simulated by the model. In accordance, electromyographic signals showed brief simultaneous inactivation of opposing muscles during movements. Thus, assuming that human movements are optimal with respect to a certain integral cost, the minimization of an absolute-work-like cost is supported by experimental observations. Such types of optimality criteria may be applied to a large range of biological movements.