The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms

Abstract

Fractal-like networks effectively endow life with an additional fourth spatial dimension. This is the origin of quarter-power scaling that is so pervasive in biology. Organisms have evolved hierarchical branching networks that terminate in size-invariant units, such as capillaries, leaves, mitochondria, and oxidase molecules. Natural selection has tended to maximize both metabolic capacity, by maximizing the scaling of exchange surface areas, and internal efficiency, by minimizing the scaling of transport distances and times. These design principles are independent of detailed dynamics and explicit models and should apply to virtually all organisms.

Some mathematical and technical issues in the measurement and interpretation of open-circuit indirect calorimetry in small animals

“…In recent years, a body of theoretical work has become established, based on fractal geometry of O 2 supply networks, explaining why 0.75 would be expected to be the exponent for the scaling of metabolic rate. 17 But this work largely ignores the fact that exponents of 0.75 have never been an established feature of the empirical database. Indeed, more recent empirical work has established that, once phylogenetic effects are accounted for (in other words, effects due to a shared evolutionary origin), the most parsimonious exponent is strongly dependent on the group being studied, [18][19][20] and there is no uniform scaling exponent -be it 0.66 or 0.75 -for either basal or maximal metabolic rate.…”

Optimality in the developing vascular system: Branching remodeling by means of intussusception as an efficient adaptation mechanism

“…A third alternative was proposed with the "West-Brown-Enquist Law" with ⌬ between 2.0 and 3.0, depending on bifurcation level. This variant additionally assumed "fractal-like" ramification patterns and made interesting predictions about the scaling properties (i.e., body size dependency) of vascular systems in various species (West et al, 1997(West et al, , 1999.…”

Structural and biophysical simulation of angiogenesis and vascular remodeling

“…Moreover, the model does not make a priori assumptions about a "fractal" geometry (West et al, 1997(West et al, , 1999 but still can lead to apparently self-similar bifurcation patterns in some regions, which correspond closely to the vascular patterns in real, rapidly growing, and remodeling networks . We wish to emphasize that statements such as "blood vessels are fractals" fail to address properly both the underlying mathematics for generating fractals and for measurement of a fractal dimension (Kurz, 2000;Sandau and Kurz, 1997).…”

“…We wish to emphasize that statements such as "blood vessels are fractals" fail to address properly both the underlying mathematics for generating fractals and for measurement of a fractal dimension (Kurz, 2000;Sandau and Kurz, 1997). Notably, fundamental aspects of transport networks have been referred to as merely "fractal-like" (West et al, 1999) and, as we proposed earlier (Kurz and Sandau, 1998), it has been shown that fractal geometry is not necessary for the observed scaling relations across animal species (Banavar et al, 1999). Although the issue of "fractal ge- ometry" is a matter of ongoing debate, we believe that a detailed analysis of biophysics, including transport of oxygen and metabolites, should always be aware of "nonfractal" topographical restrictions and morphologic heterogeneity present in vascular systems.…”

“…Several modeling approaches have been introduced before to better understand transport phenomena in both normal Sandau and Kurz, 1994) and neoplastic growth (Baish et al, 1996;Chaplain and Anderson, 1996;Chaplain and Sleeman, 1990). Moreover, computer simulations of transport systems in general have led to far-reaching conclusions about the design limits of living structures (Kurz andSandau, 1997, 1998;LaBarbera, 1990;Sandau and Kurz, 1994;Spatz, 1991;West et al, 1997West et al, , 1999.…”