Models for growth, decline and regrowth of the dendrites of rat Purkinje cells induced from magnitude and link-length analysis.

This study examines Purkinje neurons of rats aged 1, 10, 18 and 28 months to investigate growth and decline in the magnitude of the dendritic tree, i.e. the number of exterior links (terminal segments) per cell. Growth in the mean number of exterior links was observed from 1 to 10 months, decline at 18 months and regrowth at 28 months. At 10, 28, and especially at 18 months, the cell size frequency distribution indicates two groups of cells, one of small and the other of large sized cells. The study also examines the relationship of age to lengths of topologically defined links of various types. For each age group we find that exterior links are longer than interior links (non-terminal or intermediate segments). Analysis of the geometric mean lengths of subtypes of exterior and interior links at maturity (10 months) indicates that they follow a Fibonacci series of link lengths, such that mean lengths of topologically defined types of mean exterior links are either about 13 or 8 microns long, while interior links are about 5 microns long. A sequential growth model for adding exterior links is suggested to illustrate a style of growth which could account for the various mean link lengths and the Fibonacci ratio (1.618) between their lengths. Interior link lengths are also dependent on the generation of exterior links from the sides of pre-existing interior links. If the Strahler branching ratio, Rb, should increase owing to growth of terminals from interior links, then mean interior link length would decline. During a period of regression, mean exterior link lengths become shorter and mean interior link lengths become longer. Changes in mean interior link length are much less affected by changes in Rb during regression than is the situation during growth. Finally, the changes in link lengths dictate that the ratio of mean exterior to mean interior link length increases during growth phases from 1 to 10 and 18 to 28 months, and declines during regression from 10 to 28 months. The lowest values of the ratio of mean exterior to mean interior lengths are found at 1 month. This is the period of most intense growth. During this period, the rate of development of new exterior links outbalances the rate at which the links increase in length.

Tree asymmetry--a sensitive and practical measure for binary topological trees.

The topological structure of a binary tree is characterized by a measure called tree asymmetry, defined as the mean value of the asymmetry of its partitions. The statistical properties of this tree-asymmetry measure have been studied using a growth model for binary trees. The tree-asymmetry measure appears to be sensitive for topological differences and the tree-asymmetry expectation for the growth model that we used appears to be almost independent of the size of the trees. These properties and the simple definition make the measure suitable for practical use, for instance for characterizing, comparing and interpreting sets of branching patterns. Examples are given of the analysis of three sets of neuronal branching patterns. It is shown that the variance in tree-asymmetry values for these observed branching patterns corresponds perfectly with the variance predicted by the used growth model.

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.

Relation of branching angles to optimality for four cost principles.

The literature has suggested that branching angles depend on some principle of optimality. Most often cited are the minimization of lumen surface, volume, power and drag. The predicted angles depend on the principle applied, chi and alpha. Assuming flow o r chi, chi can be determined from r chi 0 = r chi 1 + r chi 2 when the radii of the parent (r0) major (r1) and minor (r2) daughters are known. The term alpha = r2/r1. Using different values for chi and alpha, we present graphs for the major and minor branching angles theta 1 and theta 2 and psi = theta 1 + theta 2 for each of the four optimization principles. Because psi is almost independent of alpha for values of chi and alpha found in 198 junctions taken from a human pulmonary artery, we are able to produce a plot of psi versus chi for each of the four principles on one graph. A junction can be provisionally classified as optimizing for a given principle if, knowing chi, the psi obs - psi pred is least for that principle. We find that this nomographic classification agrees almost perfectly with a previous classification based on a more exacting measure, the percent cost index I, where I = observed cost/minimum cost. We explain why this is to be expected in most but not all cases. First we generate a contoured percent cost surface of c = I - 100 around the optimally located junction, J, and superimpose a surface of equal angular deviations a = psi pred-psi obs. We find that c increases and a usually increases with distance from J as the actual junction moves along a straight line away from J. We then produce a plot of c versus a for two competing principles. A comparison of the principles demonstrates that, for most cases, a is smaller for the principle which has the smaller c value.

Comparison of vertex analysis and branching ratio in the study of trees.

Many kinds of naturally occurring trees have been the subject of study by investigators from a wide variety of disciplines, employing different techniques each with its own advantages and disadvantages. In this paper two such techniques for studying trees are compared: one is the classification of branches by order and the calculation of the branching ratio; the other is vertex analysis. The two methods, which at first sight appear different, are found to be mathematically similar. In complete trees, much the same information can be obtained from counting branches in each order as can be obtained from vertex analysis. In the case of pruned trees, overall branching ratio may give the more consistent results.

Branching ratio and growth of tree-like structures.

Dichotomously branching trees were generated by computer using random terminal and random segmental growth. The branching ratio (Rb) of such a tree during growth oscillates periodically as new branches are added. The magnitude of the oscillations diminishes as the tree enlarges and Rb converges towards an expected value. This phenomenon was investigated using the reverse of the growth process, that is by terminal or segmental subtraction of branches from existing trees. These were either computer generated trees or mammalian bronchial tree data. The oscillations of Rb thus obtained were similar to those obtained during growth and were used to calculate convergent values of Rb. In addition, an estimate of convergent Rb was obtained from the mean of the maximum and minimum Rb of the first oscillation occurring when the least number of branches had been subtracted. Values of Rb obtained by these methods were compared with those obtained by taking the antilogarithm of the slope of the regression of log number of branches against order. With large trees the results are similar, but with smaller trees a more reliable Rb is given by the means of the oscillations. We find that Rb values from the bronchial trees are different from those generated by random segmental growth and are not always in good agreement with random terminal growth. Some other growth process must therefore be operative in the bronchial tree.

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 alpha (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.