Viscosity can be determined from measurements of absorption of sound by utilizing the equation [A1 1]: 

 = 2p c3g (A . 3) 



where a is the absorption and u> the circular frequency. Many measurements of absorption in tissue have 

 been made at high frequencies. However, tissue exhibits relaxation phenomena at high frequencies so that 

 viscosities at frequencies above relaxation cannot be directly related to those below and can be used only 

 as lower limits. It has been found that for muscle, n s relaxes near 400 kHz and n b relaxes near 40 kHz 

 [A12]. Only one set of absorption measurements has been conducted below the Megahertz range. These 

 measurements were made at 300 to 350 kHz [A1 3]. Utilizing equation A-3, E; was calculated from these data 

 to range from 30 to 220 poise, with an average value of 130 poise. However, r| b has already relaxed at the 

 measurement frequencies, so that its contribution to the absorption is unknown. Thus, equation A-3 

 provides a lower limit to E|. An upper limit can be determined for E; if it is assumed that the absorption is due 

 solely to n s - Then, at lower frequencies, Ej would range from 1 30 to 950 poise, with an average value of 580 

 poise. Thus, from the absorption measurements: 



30 poise < E, < 950 poise. 



Viscosity can also be estimated from measurements of complex shear modulus by utilizing the equation 

 [A9]: 



n * -"ST ' (A-4) 



where u, is the imaginary part of the complex shear modulus. One set of measurements of the complex 

 shear modulus of fish tissue has been made from about 2 to 14 kHz [27]. Utilizing equations A-4 and A-1 , E, 

 was calculated from these data to range from about 4 to 90 poise, with an average value of 28 poise. 



The viscosity of animal tissue can also be estimated from the viscosity of animal cells if it is assumed 

 that the tissue viscosity is equivalent to the cell viscosity. Separate measurements have been made on the 

 viscosities of both cell protoplasm and membrane. Thus, to estimate the viscosity of the complete cell, a 

 geometric average of the membrane and protoplasmic viscosities is calculated based on the proportional 

 thicknesses of membrane and protoplasm. The equation used to calculate cell viscosity is 



f = r a ( b ~ a ) ef r 



S: [ b2 «ipSm J • (A-5) 



where E; c , E; p and Ej m are the viscosity parameters of the cell, protoplasm and membrane, respectively, and a 

 and b are the inner and outer membrane radii. Cell radii range from 2 x 10 -4 to 15 x 10 4 cm and cell 

 membranes are 75 x 10" 8 to 10~ 6 cm thick [A14]. Thus 



5 x 10- 4 < k-I 3 - <5 x 10- 3 , 

 b 



and a/b£ 1 . Measurements have been made on r\ s of protoplasm and r\ b of membranes. For protoplasm, r\ s 



was found to range from 4 x 10~ 2 to 3 x 10- 1 poise [A15]. For membranes, r\ b was found to range from 2.7 



x 10 7 to 2.7 x 10 8 poise [A16]. Therefore, utilizing equations A-2 and A-5, 



60 poise < E, c < 1,600 poise. 



This estimate is probably subject to the greatest error of the three indirect estimates of viscosity due to all 

 the assumptions required. 



Summarizing the ranges of E, determined by the various methods in their probable order of accuracy: 



direct measurement, 



430 poise < E, < 1,800 poise; 

 absorption, 



30 poise < E, < 950 poise; 

 complex shear modulus, 



4 poise < E; < 90 poise; 

 cell viscosity, 



60 poise < E; < 1 ,600 poise. 



38 



