Similarity Analysis of Physiological Systems

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SIMILARITY ANALYSIS OF PHYSIOLOGICAL SYSTEMS WALTER R. STAHL, M.D.* I. Introduction The comparative study of physiological systems in related organisms leaves little doubt that these systems function in a similar manner. For example, the lungs ofa mouse or steer are quite similar in tissue architecture and may be studied by the use ofthe same physiological variables, such as flow rate, frequency, compliance, resistance, and so forth. It may be asserted that mammalian lungs are models ofeach other and that they can be analyzed by use of modeling methods based on dimensionless numbers (similaritycriteria), in the same manneras thishas been done in die physical sciences. A major result ofsuch an analysis is that it should be possible to trace changes in system design among genetically related organisms in a quantitative manner. The similarity approach may be illustrated with a typical physical example . Suppose tliat a new aircraft design concept is to be tested by the use of a small-scale model, which is precisely similar in geometric measurements to the proposed full-size device. Ifthis model is studied in a wind tunnel in order to obtain precise data on lift, drag, and other forces, one must have some specific guide for choice of the air velocity in the wind tunnel. Stated more precisely, the modelmust be tested under conditions ofphysical similarity with the prototype. The simplest condition for physical similarity is that the model be geometrically similar and be operated so that the Reynolds No. is constant for the model and prototype. * Oregon Regional Primate Research Center and Department of Mathematics, Oregon State University. This work was completed while the author was a Special Post-Doctoral Fellow ofthe National Institutes ofHealth (NINDB BT-355). The writer wishes to acknowledge the helpful comments and assistance of Professor H. Goheen, of the Department of Mathematics, Oregon State University, and Dr. Lee Lusted, Chairman, Department ofAutomatic Data Processing and Mathematics , Oregon Regional Primate Research Center. 291 The Reynolds No. is named for Osborne Reynolds, who cited it in 1883 in an historic paper entitled "An Experimental Investigation of the Circumstances Which Determine Whether the Motion ofWater Shall Be Direct or Sinuous, and ofdie Law ofResistance in Parallel Channels" [1]. It has the following composition: vLp/v(1) in which ? = flow velocity (L/T); L = some characteristic length ofthe system; ? = density (M/Ls) and ij = viscosity (M/LT). Reynolds apparently obtained this similarity criterion empirically, but it is easily derived from the Navier-Stokes equations of hydrodynamics as is shown in Langhaar [2]. The particular significance of the Reynolds No. is that it is a ratio of inertial to viscous forces in a system involving a flowing fluid. For many of the simpler cases ofhydrodynamic modeling, constancy of this dynamic ratio is a sufficient condition forphysical similarity. The numericalmagnitude oftheReynoldsNo. indicateswhetherinertialorviscous forces predominate. Ifthe inertial forces are greatly in excess (numerical value over 2,000, approximately), the system is no longer held together by viscous forces and turbulence will occur. Some 80 or more dimensionless numbers are cited in the literature on dimensional and similarity analysis. Among the more important reference works on the subject are volumes by Bridgman [3], Langhaar [2], Duncan [4], Focken [5], Birkhoff [6], and Sedov [7]. Similarity and dimensional methods have been applied successfully to all types ofengineering problems , particularly those having to do with fluid flows, heat transfer (McAdams [8]), magneto-hydrodynamics (Cowling [9]), rheology (Eirich [10]), chemical engineering simulation (Johnstone and Thring [11]), and numerous other special areas. In biology there have been many prior attempts to make use of the concept of similarity, but the present writer was apparently the first to attempt a systematic study of biologically significant similarity criteria (dimensionless numbers) with full application of the modern techniques ofsimilarity analysis [12-15]. m die materials given below, the constancy ofspecific similarity criteria for certain biological systems is demonstrated, and representative analyses are provided for complexphysiological systems and artificial organs, with use of equations relating several similarity criteria . 292 Walter R. Stahl · Similarity Analysis Perspectives in Biology and Medicine · Spring 1963 II. Mathematical Basis ofSimilarity Theory The application of dimensional methods to biology requires a clear understanding ofsimilarity theory. The problems in biology are as...