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WALTEE R. STAHL

numerical or Computer analyses based on physical eąuations which imply physical similarity critera; and (5) miscellaneous devices, such as the artificial kidney, which are partial models and only qualitatively similar to their real prototype.

This chapter will not attempt to cover electroanalog model theory in detail. An excellent reference on this problem is the volume of Murphy and associates (1963), which deals with both physical and electrical analogs, and reveals the isomorphisms between these two methods. A very complete survey of available literaturę on electroanalogical methods has been provided by Higgins (1957). The use of electrical analogs in biology is discussed by Dainty (1960), who presents examples based on heat flow, enzyme kinetics, and electrical membranę theory. Dainty does not, however, discuss the fact that for quantitative extrapolation the dimensionless numbers in the electrical system must correspond to the natural goveming physical similarity criteria of the prototype. Analog models have also been used extensively for simulation of physio-logical control; in this instance, criteria other than equivalent physical dimensionless numbers are the pertinent invariants, as discussed below.

The generał question of physical models in biology has been discussed by Beament (1960, pp. 83-101), but with very little use of physical similarity theory and dimensionless numbers. He stresses problems of finding suitable modeling materials, the present inability to simulate certain molecular properties, and other practical modeling problems. In the same symposium volume on biological models, R. H. J. Brown (1960) considers “mechanical models in zoology,” that are taken to include simple physical machines such as levers or wheels. Interesting models of animal locomotion and bird flight may be obtained in this manner and are discussed in morę detail in a later section. Brown men-tions some scalę effects and he also refers to size-independent aero-dynamic performance criteria in analysis of bird flight. Insofar as the writer is aware, there have been no prior published discussions covering the entire problem of similarity criteria for physical analog models of organisms, including also artificial organs, but a great many specific reports dealing with physical analogs of living systems are available.

A. Cardiovascular System Analogs

The most obvious area for physical simulation is probably the cardio-yascular system, to which one can apply certain existing principles of hydrodynamic modeling. A number of model experiments are cited in the comprehensive discussion of hemodynamics by McDonald (1960). On the basis of work of Taylor and Womersley, McDonald concludes