86 



HANDBOOK OF PHYSIOLOGY 



CIRCULATION I 



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FIG. I. The variety of sizes and composition of the wall of 

 the difTerent blood vessels. [From Burton (6).] 



through noiirigid, distensible blood vessels are dis- 

 cussed elsewhere in this volume and in the literature 

 (15). Poiseuille's law is that the flow F is given by: 



F = aP 



x(Ox(OKO 



(0 



where AP is the pressure difference (i.e., the driving 

 force), (tt/S) is the "numerical factor" arising from 

 the integration over a cross section of a cylindrical 

 tube in the development by Hagen, (i/??) is the 

 "viscosity factor," and (rVO is the "geometrical 

 factor" involving the values r and the length / of the 

 vessel concerned. By definition, the resistance to flow 

 is the ratio of the driving force, i.e., the pressure 

 difference, to the flow that results. 



AP 



■e) 



X 7, X 



02) 



The resistance then depends on the two factors, the 

 viscosity factor rj, and the geometrical factor (//r''). 



The "fourth power law," by which the resistance to 

 flow is inversely proportional to the fourth power of 

 the radius, gives a very sensitive control of the dis- 

 tribution of blood flow by the caliber of the resistance 

 v'essels, i.e., the arterioles. A reduction in radius of 

 only 16 per cent will double the resistance, and of 



50 per cent will increase the resistance bv 2'', i.e., 

 16 times. 



For a fluid, e.g., water, which obeys Newton's law 

 of viscosity, and thus is a "Newtonian fluid," the 

 coefficient of viscosity jj is independent of the rate of 

 flow or of the size of the vessel. Blood, however, is a 

 complicated fluid containing large particles (e.g., the 

 erythrocytes), is not Newtonian in its viscosity, and 

 t) varies with the rate of flow (axial accumulation) 

 and with the size of vessels (F&hraeus-Lindqvist 

 effect) (16). For explanation of these see the following 

 chapter. It has been shown that in spite of this, in 

 the physiological range of blood flow, blood behaves 

 very accurately as if it were Newtonian (15). 



2. importance of the distensibility of blood 

 vessels: transmural pressures 



The factor of more importance, in the Poiseuille 

 equation, which can alter the resistance and the dis- 

 tribution of blood in the body is thus the "geometrical 

 factor" (//V''). In rigid tubes, as used by Poiseuille, 

 this factor is a constant. Blood vessels are distensible, 

 so that their geometry depends upon the pressure 

 within them. The particular pressure concerned is, 

 for each vessel, its "transmural pressure," Pt.v , i.e., 

 the difference of pressure from inside, where the 

 pressure is the "intravascular pressure," P,y , to the 

 outside, where the pressure is the "tissue pressure," 

 Pt , i.e., 



Piv — Pt — Ptm 



(3) 



While in equation 2 the resistance R does not 

 depend explicitly upon the pressure difference AP, 

 nevertheless where the vessels are distensible the 

 resistance will depend upon AP. This is because any 

 change in AP will usually involve a change in the 

 transmural pressure, Ptm, of the "resistance vessels." 

 For example, the usual way to produce a flow- 

 pressure curve would be to keep the venous pressure 

 constant, and change the arterial pressure to different 

 values. This will inevitably change the transmural 

 pressures in all the vessels between artery and vein, 

 so their geometrical factors (//r*) will be changed. 



In another, quite different, experiment, to verify 

 equation i we might increase the driving pressure 

 AP by keeping the arterial pressure constant, but 

 reducing the venous pressure. In this case the trans- 

 mural pressure of all the vessels would decrease as 

 AP was increased (whereas in the first case it would 

 increase) and the effect on the geometry of the vessels 



