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THE ROLE OF MODELS IN THEORETICAL BIOLO"tY 177

that considerable useful information can be obtained from models about such matters as pulsatile blood flow in elastic tubes, reflection of pulse waves at complex vascular bifurcations, effects of viscosity on pulsatile flow, etc. He makes it elear, however, that the models used to datę are very imperfect and usually provide only semiquantitative information about some one process among the many involved in cardiovascular system function in situ. Several specific dimensionless numbers are used by McDonald as design criteria for vascular models.

A large number of reports on cardiovascular models are available. Ostrach (1964) has used an elastic tubing model driven by pulsatile flow to study pulse wave effects. He has been able to demonstrate certain phenomena seen in animals, e.g., pulse wave peaking and reflection, in a satisfactory manner and lists three dimensionless numbers as similarity criteria relating his model and mammals. A similar study on rigid tubes was reported by Coulter and Kunz (1964), who were able to confirm certain relationships between flow impedance and freąuency proposed by Womersley for animal vascular systems. Attinger et al. (1964) have reported on an elastic tubę model driven by pulsating flow that is similar in size to a dog aorta and invariant under several pertinent similarity criteria, including a modified Reynolds number. Transparent models of the aorta filled with flowing bentonite suspension were used as models of the aorta in the work of Meisner and Rushmer (1963). Very complex flow patterns and turbulence were found to occur at comparatively Iow Reynolds numbers near simulated valves and aortic constructions.

A geometrically precise, but eight times enlarged, physical model of the circle of Willis (a complex flow juncture in the brain) was created by M. E. Clark et al. (1963), on the assumption of invariance of both the Reynolds and Euler numbers. This reąuired use of a fluid different from blood in the model. The results obtained indicated extremely complex flow patterns in this ringlike vascular junction, which probably cannot be predicted by analytic means, particularly for pulsatile flow.

A mixed natural-artificial modeling system is exemplified by work of Tempie et al. (1964), who uses a heart and aorta obtained from autopsy to study flow patterns at the aortic valve. A cannula was inserted through the ventricular wali and a pump used to simulate cardiac action, with water as the fluid and dye injected to reveal the flow pattem. This model is based on identical geometry and constancy of the Euler number, but has a Reynolds number which is four times too high because water is about one-fourth as viscous as blood.

For some years Burton (1965) and Prothero and Burton (1962) have studied smali tubing models of blood flow based on the Poiseuille number,