Nature’s Patterns and the Fractional Calculus by Bruce J. West

Author:Bruce J. West
Language: eng
Format: epub, pdf
Publisher: Walter de Gruyter
Published: 2017-06-15T00:00:00+00:00

4.2.3The optimum can be dangerous

A question of interest is whether optimal design criteria are ever realized in nature and whether they are even desirable? The WBE model suggests that the AR exponent value of 3/4 is proof of the optimally of fractal design of physiological networks. However, the controversy over the empirical value of the allometry exponent calls the applicability of the ‘proof’ into question, which is to say, although the mathematics cannot be faulted, its applicability to a wide range of physiological networks remains doubtful.

Another distinct application of the fractal design principle was made to the mammalian lung by West et al. [338], who established that the average diameter of a bronchial tube, as a function of generation number, is described by a modulated IPL. This discussion regarding the mammalian lung and whether the bronchial tree is optimal has been investigated by Mauroy et al. [208]. The latter authors maintain that the mammalian bronchial tree is a good example of an efficient distribution network, with an approximate fractal structure [230, 338]. They state that physical optimization is critical in that small variations in the geometry can induce large variations in the net air flux and consequently, optimality cannot be a sufficient criterion for physiologic design of the bronchial tree. The slight deviations observed in the parameters presumed to be optimized are a manifestation of a safety factor that has been incorporated into the design and hence into the capacity for regulating airway caliber.

In the present context the size ratio h of successive airway segments are homothetic with h = 2−1/3 ≈ 0.79, as discussed earlier. Homothetic scaling means that the lengths and diameters have the same ratios, between successive generations. Using the resistance minimization argument for the bronchial network Mauroy et al. [208] show that the ‘best’ bronchial tree is fractal, with constant reduction factor given by the Hess–Murray Law. Do the data support this optimal value? If not, what does that imply about the efficiency of bronchial airways?

The fractal dimension for a bronchial airway is D = − ln 2/ln h so that the Hess–Murray Law implies D = 3, whereas h > 0.79 implies D > 3. In the human lung it is found that the homothety ratio is h ≈ 0.85 [332] and the bronchial network is therefore not optimized: its volume is too large and its overall resistance is too small. Mauroy et al. [208] emphasize that this deviation from optimal is, in fact, a safety margin for breathing, with respect to possible bronchial constrictions.

Sapoval [281] has argued that without regulation of the airway caliber [259], there would be a multifractal spatial distribution of air within the lungs, resulting in strongly non-uniform ventilation, with some regions of the lung being poorly fed with fresh air. Expanding on this theme, using inhomogeneity of the homothety ratio, Mauroy et al. [208] show how the optimal network is dangerously sensitive to physiological variability and consequently effective design of the bronchial tree must incorporate more than just physical optimality. This argument has clear implications for other scaling networks as well.

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