Similarity and Dimensional Methods in Biology

Abstract

A review is provided of the modern interpretations of "dimensions" and dimensional analysis. It is shown that recent work has put dimensional analysis on a very firm mathematical basis which relates to the group theory of transformations in advanced algebra. The concept of biological similarity, or more specifically that of physical similarities between genetically related organisms of varying size, is examined historically and found to have many antecedents, particularly in the work of D'' Arcy Thompson. Quantitative allometric (power law scaling of a function against body mass) data is combined into non-dimensional criteria and also into dimensional constants which are shown to be independent of the mass of the organism. It is proposed that such invariants show quantitatively in what sense a small organism is a model of a large one. This demonstration is based on the fact that it is possible to cancel out the exponents in allometric laws by combining them into various ratios, involving 2-3 or more components. The actual allometric data used in the report is based on a wide variety of accepted published findings. Consideration is also given to the problem of making models of biological systems which are physically similar (true models) to the natural system. The artificial kidney is found to be a limited model of the natural nephron, but does show some important and basic functional similarities that are interpreted by use of suitable dimensionless numbers An example of a completely new dimensionless number which is of proven utility for analysis of the cardiovascular and respiratory systems is offered. Comments are provided on the relations of biological scaling laws and the scaling methods which have come into use in chemical engineering practice. The remarkably wide range of sizes in which a mammalian physiological system may function is discussed and found to be the result of rigorous control of a great many physical variables, such as viscosity or blood pressure, which allows scaling up on the basis of constancy of comparatively few physical similarity requirements. An extensive bibliography of dimensional analysis of biological similarity is included.