Relation of branching angles to optimality for four cost principles

Morphometrics predicts the differential regurgitant fraction in bilateral pulmonary arteries of patients with repaired tetralogy of fallot

In patients with repaired tetralogy of Fallot (rTOF), the regurgitant fraction (RF) in left pulmonary artery (LPA) and right pulmonary artery (RPA) is usually unequal. The morphometrics may play a crucial role in this RF discrepancy. Cardiovascular MR of 79 rTOF patients and 20 healthy controls were retrospectively enrolled. Forty-four from the 79 patients were matched in age, sex and body surface area to the 20 controls and were investigated for: (1) phase-contrast flow of main pulmonary artery (MPA), LPA, and RPA; (2) vascular angles: the angles between the thoracic anterior-posterior line (TAPL) with MPA (θM-AP), MPA with RPA (θM-R), and MPA with LPA (θM-L); (3) cardiac angle, the angle between TAPL and the interventricular septum; (4) area ratio of bilateral lung and hemithorax regions. Compared with the 20 controls, the 44 rTOF patients exhibited wider θM-AP, sharper θM-L angle, and a smaller θM-L/θM-R ratio. In the 79 rTOF patients, LPA showed lower forward, backward, and net flow, and greater RF as compared with RPA. Multivariate analysis showed that the RF of LPA was negatively associated with the θM-L/θM-R ratio and the age at surgery (R2 = 0.255). Conversely, the RF of RPA was negatively associated with the left lung/left hemithorax area ratio and cross-sectional area (CSA) of LPA, and positively associated with CSA of RPA and MPA (R2 = 0.366). In rTOF patients, the RF of LPA is more severe than that of RPA, which may be related to the vascular morphometrics. Different morphometric parameters are independently associated with the RF of LPA or RPA, which may offer potential insights for surgical strategies.

Fractal analysis of concurrently prepared latex rubber casts of the bronchial and vascular systems of the human lung

Fractal geometry (FG) is a branch of mathematics that instructively characterizes structural complexity. Branched structures are ubiquitous in both the physical and the biological realms. Fractility has therefore been termed nature's design. The fractal properties of the bronchial (airway) system, the pulmonary artery and the pulmonary vein of the human lung generates large respiratory surface area that is crammed in the lung. Also, it permits the inhaled air to intimately approximate the pulmonary capillary blood across a very thin blood-gas barrier through which gas exchange to occur by diffusion. Here, the bronchial (airway) and vascular systems were simultaneously cast with latex rubber. After corrosion, the bronchial and vascular system casts were physically separated and cleared to expose the branches. The morphogenetic (Weibel's) ordering method was used to categorize the branches on which the diameters and the lengths, as well as the angles of bifurcation, were measured. The fractal dimensions (DF) were determined by plotting the total branch measurements against the mean branch diameters on double logarithmic coordinates (axes). The diameter-determined DF values were 2.714 for the bronchial system, 2.882 for the pulmonary artery and 2.334 for the pulmonary vein while the respective values from lengths were 3.098, 3.916 and 4.041. The diameters yielded DF values that were consistent with the properties of fractal structures (i.e. self-similarity and space-filling). The data obtained here compellingly suggest that the design of the bronchial system, the pulmonary artery and the pulmonary vein of the human lung functionally comply with the Hess-Murray law or 'the principle of minimum work'.

THE OPTIMAL FORM OF DISTRIBUTION NETWORKS APPLIED TO THE KIDNEY AND LUNG

A model is proposed to minimize the total volume of the main distribution networks of fluids in relation to the organ form. The minimization analysis shows that the overall exterior form of distribution networks is a modified ellipsoid, a geometric form that is a good approximation to the external anatomy of the kidney and lung. The variational procedure implementing this minimization is similar to the traditional isoperimetric theorems of geometry.

A novel measure to characterise optimality of diameter relationships at retinal vascular bifurcations

Conventionally, the relationship between parent and daughter vessels at vascular bifurcations has been expressed by the junction exponent (x), and deviations of this parameter from the optimal conditions predicted by Murray’s law (x = 3) have been shown to be associated with vascular disease. However, the junction exponent is normally calculated iteratively from diameter measurements, and Monte-Carlo simulation studies show the junction exponent to be biased in the presence of measurement noise.

We present an alternative parameter, referred to as optimality ratio, that is simpler to compute and also more robust in the presence of noise.

To demonstrate the sensitivity of the optimality ratio to alterations in topography of the retinal vascular network, we analysed the effect of inducing endothelial dysfunction by infusion of NG-monomethyl-l-arginine (l-NMMA), a nitric oxide synthase inhibitor, compared to placebo in a double-blind crossover study. The optimality ratio showed a significant increase (p = 0.03) during infusion of l-NMMA compared to placebo.

We propose that a measure of the extent of departure of optimality ratio from its optimal value of 2^−1/3 may be a useful indicator of microvascular endothelial dysfunction in vivo.

Digital image analysis of plus disease in retinopathy of prematurity

An accurate assessment of retinopathy of prematurity (ROP) is essential in ensuring correct and timely treatment of this potentially blinding condition. Current modes of assessment are based upon clinical grading by expert examination of retinal changes. However, this may be subjective, unreliable and difficult and there has been significant interest in alternative means of measurement. These have been made possible through technological advancements in image capture and analysis as well as progress in clinical research, highlighting the specific importance of plus disease in ROP. Progress in these two fields has highlighted the potential for digital image analysis of plus disease to be used as an objective, reliable and valid measurement of ROP. The potential for clinical and scientific advancement through this method is argued and demonstrated in this article. Along with the potential benefits, there are significant challenges such as in image capture, segmentation, measurement of vessel width and tortuosity; these are also addressed. After discussing and explaining the challenges involved, the research articles addressing digital image analysis of ROP are critically reviewed. Benefits and limitations of the currently published techniques for digital ROP assessment are discussed with particular reference to the validity and reliability of outcome measures. Finally, the general limitations of current methods of analysis are discussed and more diverse potential areas of development are discussed.

Retinal image analysis: concepts, applications and potential

As digital imaging and computing power increasingly develop, so too does the potential to use these technologies in ophthalmology. Image processing, analysis and computer vision techniques are increasing in prominence in all fields of medical science, and are especially pertinent to modern ophthalmology, as it is heavily dependent on visually oriented signs. The retinal microvasculature is unique in that it is the only part of the human circulation that can be directly visualised non-invasively in vivo, readily photographed and subject to digital image analysis. Exciting developments in image processing relevant to ophthalmology over the past 15 years includes the progress being made towards developing automated diagnostic systems for conditions, such as diabetic retinopathy, age-related macular degeneration and retinopathy of prematurity. These diagnostic systems offer the potential to be used in large-scale screening programs, with the potential for significant resource savings, as well as being free from observer bias and fatigue. In addition, quantitative measurements of retinal vascular topography using digital image analysis from retinal photography have been used as research tools to better understand the relationship between the retinal microvasculature and cardiovascular disease. Furthermore, advances in electronic media transmission increase the relevance of using image processing in 'teleophthalmology' as an aid in clinical decision-making, with particular relevance to large rural-based communities. In this review, we outline the principles upon which retinal digital image analysis is based. We discuss current techniques used to automatically detect landmark features of the fundus, such as the optic disc, fovea and blood vessels. We review the use of image analysis in the automated diagnosis of pathology (with particular reference to diabetic retinopathy). We also review its role in defining and performing quantitative measurements of vascular topography, how these entities are based on 'optimisation' principles and how they have helped to describe the relationship between systemic cardiovascular disease and retinal vascular changes. We also review the potential future use of fundal image analysis in telemedicine.

Retinal vascular image analysis as a potential screening tool for cerebrovascular disease: a rationale based on homology between cerebral and retinal microvasculatures

The retinal and cerebral microvasculatures share many morphological and physiological properties. Assessment of the cerebral microvasculature requires highly specialized and expensive techniques. The potential for using non-invasive clinical assessment of the retinal microvasculature as a marker of the state of the cerebrovasculature offers clear advantages, owing to the ease with which the retinal vasculature can be directly visualized in vivo and photographed due to its essential two-dimensional nature. The use of retinal digital image analysis is becoming increasingly common, and offers new techniques to analyse different aspects of retinal vascular topography, including retinal vascular widths, geometrical attributes at vessel bifurcations and vessel tracking. Being predominantly automated and objective, these techniques offer an exciting opportunity to study the potential to identify retinal microvascular abnormalities as markers of cerebrovascular pathology. In this review, we describe the anatomical and physiological homology between the retinal and cerebral microvasculatures. We review the evidence that retinal microvascular changes occur in cerebrovascular disease and review current retinal image analysis tools that may allow us to use different aspects of the retinal microvasculature as potential markers for the state of the cerebral microvasculature.

On the principles of the vascular network branching

We propose an explanation of Murray's law without applying the minimality principles. The model deals with a "delivering" artery system of an organ that is characterized, first, by the space-filling embedding into the organ tissue and, second, by the uniform distribution of the blood pressure drop over it. The latter assumption is justified using the available physiological data and the idea about conditions needed for perfect self-regulation. Based on the two statements we get Murray's law, and so, demonstrate that it can be also regarded as a direct consequence of the organism's capacity for controlling finely the blood flow redistribution over peripheral vascular networks.

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.

Modeling the large-scale geometry of human coronary arteries

Two principles suffice to model the large-scale geometry of normal human coronary arterial networks. The first principle states that artery diameters are set to minimize the power required to distribute blood through the network. The second principle states that arterial tree geometries are set to globally minimize the lumen volume. Given only the coordinates of an arterial tree's source and "leaves", the model predicts the nature of the network connecting the source to the leaves. Measurements were made of the actual geometries of arterial trees from postmortem healthy human coronary arteriograms. The tree geometries predicted by the model look qualitatively similar to the actual tree geometries and have volumes that are within a few percent of those of the actual tree geometries. Human coronary arteries are therefore within a few percent of perfect global volume optimality. A possible mechanism for this near-perfect global volume optimality is suggested. Also, the model performs best under the assumption that the flow is not entirely steady and laminar.Key words: arteries, optimization, volume, power, geometry.

Blood flow and optimal vascular topography: role of the endothelium

We have used x-ray microangiography to investigate the influence of EDRF and endothelin-1 on arterial diameters (70-800 microns) at bifurcations in the isolated rabbit ear and the "optimality" of its branching geometry. The median value of the junction exponent x (which is given by d0x = d1x + d2x, where d0, d1 and d2 are parent and daughter artery diameters respectively) was close to 3 at different flow rates in unconstricted preparations. When x = 3, branching geometry is optimal in that i) power losses and intravascular volume are both minimised, and ii) fractal considerations suggest that the total surface area for metabolic exchange is maximised. Under conditions of vasoconstriction (by 5HT/histamine) the junction exponent deviated from its control value but was restored towards 3, both by basal and by acetylcholine-stimulated EDRF activity. In contrast, endothelin-1 caused a dose-dependent reduction in the junction exponent from its optimal value 3. This suggests that the endothelium helps to optimise microvascular function through EDRF but not endothelin-1 release.

Basal EDRF activity helps to keep the geometrical configuration of arterial bifurcations close to the Murray optimum

We have used X-ray microangiography to investigate the hypothesis that the potent endogenous vasodilator endothelium-derived relaxing factor (EDRF) contributes to the maintenance of "optimality" in vascular branching by modulating the diameters of the parent (D0) and daughter (D1 and D2) arteries at bifurcations. Five anatomically different types of bifurcation were studied in buffer-perfused rabbit ear preparations both under resting conditions and after pharmacological constriction by 5-hydroxytryptamine (5HT). A range of flow rates (1-5 ml min-1) was employed as release of EDRF from endothelial cells is stimulated by shear stress. Experimental data obtained in the presence and absence of EDRF activity were compared with theoretical predictions in three ways. (1) Junction exponents (x) were determined at each bifurcation from the equation Dx1 + Dx2 = Dx0, and their frequency distributions constructed. Murray (1926a, Proc. natn. Acad. Sci., U.S.A. 12, 207-214; 1926b, J. gen. Physiol. 9, 835-841.) proposed that x will be exactly 3 if power losses and intravascular volume are minimized simultaneously. In unconstricted preparations, either in the presence or absence of EDRF activity, and in preparations constricted by 0.1 microM 5HT in the presence of EDRF activity, the modes and medians of the frequency distributions of x were found to be close to 3 at all flow rates. In contrast, in 0.1 microM 5HT-constricted preparations in the absence of EDRF activity, no single mode common to all flow rates was apparent and medians were significantly larger at all flow rates. (2) Theoretically "optimal" branching angles were derived from experimental diameter measurements using four mathematical models which minimize respectively the total surface area, total volume, total drag (shear stress) and total power losses at bifurcations (Murray, 1926b). These calculated branching angles were then compared with actual branching angles. EDRF activity was found to be necessary for accurate prediction of branching angles by the minimum volume and power loss models in 5HT-constricted but not in resting preparations. (3) For each model or "minimization principle", there is an optimal mathematical relationship between the junction exponent, x, and the angle between daughter arteries, psi 12, at a bifurcation (Roy & Woldenberg, 1982, Bull. math. Biol. 44, 349-360.) Experimentally determined values of x and psi 12 agreed closely with those predicted both by the minimum volume and the minimum power loss principles, except again in 5HT-constricted preparations in the absence of EDRF activity.(ABSTRACT TRUNCATED AT 250 WORDS)

Principles of design of fluid transport systems in zoology

Fluid transport systems mediate the transfer of materials both within an organism and between an organism and its environment. The architecture of fluid transport systems is determined by the small distances over which transfer processes are effective and by hydrodynamic and energetic constraints. All fluid transport systems within organisms exhibit one of two geometries, a simple tube interrupted by a planar transfer region or a branched network of vessels linking widely distributed transfer regions; each is determined by different morphogenetic processes. By exploiting the signal inherent in local shear stress on the vessel walls, animals have repeatedly evolved a complex branching hierarchy of vessels approximating a globally optimal system that minimizes the costs of the construction and maintenance of the fluid transport system.

Diameters and cross-sectional areas of branches in the human pulmonary arterial tree

Measurements were made of the diameters of the three branches meeting at each of 1,937 bifurcations in the pulmonary arterial tree, using resin casts from two fully inflated human lungs. Cross-sectional areas of the parent branch and of the daughter branches were calculated and plotted on a log-log plot, which showed that mean cross-sectional area increases in a constant proportion of 1.0879 at bifurcations. The mean value of the ratio of daughter branch diameters at bifurcations was 0.7849. The mean value of the exponent z in the equation flow = k (diameter(z)) was found to be 2.3 +/- 0.1, which is equal to the optimal value for minimizing power and metabolic costs for fully developed turbulent flow. Although Reynolds number may exceed 2,000 in the larger branches of the pulmonary artery, turbulent flow probably does not occur, and in the peripheral branches Reynolds number is always low, excluding turbulent flow in these branches. This finding seems to be incompatible with the observed value of z. A possible explanation may be that other factors may need to be taken into account when calculating the theoretical optimum value of z for minimum power dissipation, such as the relatively short branches and the disturbances of flow occurring at bifurcations. Alternatively, higher arterial diameters reduce acceleration of the blood during systole, reduce turbulent flow, and increase the reservoir function of the larger arteries. These higher diameters result in a lower value of z.

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.