A critical understanding of the fractal model of metabolic scaling.

The exponent of the scaling of metabolic rate with body mass has been the subject of debate for more than a century. The argument is at two levels, one concerning questions of empirical support for the exponent and the other, how to derive it theoretically. At this second level, the exponent is usually treated as the outcome of an underlying physical burden and approached as the search for a natural law emerging within energetic and geometric constraints. Recently, a model relying on fractal geometry was proposed as a general explanation for the phenomenon. In the present study, a reanalysis of the fractal model is performed to verify its validity. All the conditions that allow for the connection between the geometric proposition and the allometric exponent are evaluated, as well as the energy loss minimization procedure put forward in the model. It is demonstrated that the minimization procedure is mathematically incorrect and ill-posed. Also, it is shown that none of the connecting conditions are fulfilled. Therefore, it is concluded that the fractal model lacks self-consistency and correct statement: it relies on strong assumptions of homogeneity in morpho-physiological features among organisms instead of demonstrating them, as claimed by its authors. It is proposed that empiricists and theoreticians should rather evaluate the frameworks for addressing metabolic scaling phenomena.