FRACTAL VASCULAR GROWTH PATTERNS

Abstract

Flow distributions in the heart and lung are heterogeneous but not at all random. The apparent degree of heterogeneity increases as one reduces the size of observable elements; the fact that the dispersion of flows shows a logarithmic relation to element size says that the system is statistically fractal. The fractal characterization is a statement that the system is nonrandom and that it shows correlation. The close near neighbor correlation has as the corollary of long tailing or falloff in correlation with distance, so called spatial persistence. Correlation can be expected because flow is delivered via a branching vascular system, and so it appears that the structure of the vasculature itself contributes. Since it is also practical and efficient for growth to occur via recursive rules, such as branch, grow, and repeat the branching and growing, it appears that fractals may be useful in understanding the ontological aspects of growth of tissues and organs, thereby minimizing the requirements for genetic material.

Keywords: angiogenesis; blood flow; correlation; heterogeneity; ontogeny.

Figure 1

Family tree of the Indo-European languages, traced back to the protolanguage that existed only 6000 years ago. Splits into dialects, then distinct languages and daughter languages are only now diminishing through the vehicle of worldwide communication of the twentieth century (from Gamkrelidze and Ivanov, 1990).

Figure 2

Sea shell, halved, showing the log spiral form (from Ghyka, 1977.)

Figure 3

A small cluster of 3000 particles formed by diffusion-limited aggregation. The first 1500 points are larger, and paucity of small dots attached to the earliest deposits indicates that very few particles penetrate the into the depths of the cluster, but are caught nearer the growing terminae (from Witten and Sander, 1983).

Figure 4

Dichotomously branching fractal model. Left: basic element in which fractions γ and 1 – γ of total flow F₀ are distributed to daughter branches. Right: flow at terminal branches in network of two generations (from Glenny and Robertson, 1991).

Figure 5

Area splitting algorithm for two dimensional dichotomous branching system filling the space available. The rule is this: from a starting point, draw a line to split the area available, and proceed along the line a specific fraction of its length (stage 1). Repeat this, splitting both halves (stage 2).

Figure 6

Area-dividing branching algorithm applied to quarter circles. The patterns are strikingly dependent on this value of the fractional distance to the far boundary (figure provided courtesy of C.Y. Wang and L.B. Weissman).

Figure 7

Tissue expansion followed by vascular sprouting. In a given iteration the cells (the squares) divide and grow (one cell → 4 cells here); the matching growth function g resizes the existing vessels, then new vessels are located in regions of insufficiency and the sprouts added to the structure (from Gottlieb, 1992, with permission of IEEE).

Figure 8

Examples of simulated drainage networks grown by the self-avoiding invasion percolation method. Numerical simulations were set up using a square site-bond lattice in which random bond strengths were normally distributed lattice size is 512 × 256. Seeding was allowed from every other point along the bottom edge. In this case invasion of the lattice was allowed to continue until no further free sites were available. Strahler stream ordering was applied: streams of order four and above are shown. The substrate becomes steadily more susceptible to northward growth from a (isotropic lattice) to c (strongly anisotropic lattice). The fractal dimension d_min of the principal streams is invariant to this anisotropy and is on average 1.30 throughout (from Stark, 1991).

Figure 9

Number of cell types in organisms seems to be related mathematically to the number of genes in the organism. In this diagram the number of genes is assumed to be proportional to the amount of DNA in a cell. If the gene regulatory systems are K = 2 networks, then the number of attractors in a system is the square root of the number of genes. The actual number of cell types in various organisms appears to rise accordingly as the amount of DNA increases. The “K = 2 networks” refers to a simple system wherein the connections between neighbors in a planar array control whether an element is active or not, a kind of binary level of control that is about the simplest conceivable for a multielement system (from Kauffman, 1991).