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THE ROLE OF MODELS IN THEORETICAL BIOLOGY 171

that performance measurements on the model can be extrapolated to tbe latter. If the pertinent physical criteria are not invariant or unknown, then data from the model system is of no certain significance for the prototype, regardless of how similar the model may be to it in appearance.

For example, performance of toy race cars which look very similar to real ones or most rubber tubing models of the cardiovascular tree cannot be extrapolated to their prototypes because all of the modeling invariants are not satisfied. In the case of the toy auto, there may be perfect geometrie similarity, but not dynamie similarity in relation to the power plant. With the vascular model certain complex dimensionless numbers pertaining to elasticity, pressure, and viscosity cannot all be held invari-ant simultaneously using available materials and fabrication methods.

B. Objective Modeling Criteria

Table I summarizes various classes of models and the pertinent numerical, algorithmic, or abstract mathematical similarity invariants which govem them. A tabulation encompassing the fuli rangę of bio-logical modeling situations is not known to have been proposed before but was outlined in an earlier report (Stahl, 1965b).

The same generał rangę of criteria is applicable to modeling of a complex physical system such as an aireraft, for which one may use physical analogs, electrical analogs, numerical methods, control theory models, and generał Computer simulation techniąues. In addition, the engineer commonly uses data from previous workable prototypes and, in this sense, treats the new design as a model of previous ones. This yiewpoint will be seen to be precisely applicable to organisms when treated as models of each other, e.g., when rabbits are exposed to blast waves from atomie weapons to predict human injury under comparable conditions.

From Table I it becomes elear that the concept of a “model” is a generał one and is used in different ways. Several points should be empha-sized about the design of models. (1) A particular model represents only certain of all possible properties of a prototype; in this respect modeling is a fundamentally arbitrary undertaking. (2) The numerical or algorithmic criteria of similarity relating a model and prototype depend on the purpose for which the model is to be used and are also arbitrary. (3) There is no imiąue fundamental justification for stating that one model is a “best model,” and choice of a model is madę on utilitarian, not metaphysical, absolute grounds. (4) In generał, any one class of model