244 Mechanical Resonances of Biological Cells / 1 3 : 5 



Sections 2 and 3 were all based on the lowest possible mode being 

 excited. However, similar studies with air bubbles showed that the 

 higher modes were easier to excite than the lower ones. Similarly, 

 photographs such as those in Figure 3 suggest that this is also true for 

 biological cells ; using a higher mode decreases the values of T calculated 

 in Section 2 and of \x in Section 3. Because \x was barely high enough 

 to be comparable with protein gels, whereas 7was higher than estimated 

 by static methods, the interpretation of the optimum frequencies as 

 higher resonant modes slightly favors the interfacial- tension model. 



5. Summary 



A mathematical theory has been presented in this chapter in terms of 

 which it is possible to explain observed maxima in the rates of destruction 

 of ciliate protozoans and vertebrate erythrocytes in cavitating acoustic 

 fields. Two different types of models were considered : one a cell sur- 

 rounded by a membrane which is the seat of an interfacial tension, and 

 the other a cell surrounded by a gel-like cortex. Both models, with 

 reasonable physical constants, predict the observed resonances. The 

 analyses were first performed on highly simplified models, and then the 

 effects of the simplifications were discussed. It is impossible to choose 

 between the various models in the light of the current experimental data. 

 All agree with the available evidence. 



REFERENCES 



The following articles by the author and his co-workers discuss in more 

 detail the material presented in this chapter. 



1. Ackerman, Eugene, "Resonances of Biological Cells at Audible Fre- 

 quencies," Bull. Math. Biophys. 13: 93-106 (1951). 



2. Ackerman, Eugene, "An Extension of the Theory of Resonances of Bio- 

 logical Cells, I. Effects of Viscosity and Compressibility," Bull. Math. 

 Biophys. 16: 141-150 (1954). 



a. "II. Cross-Section in a Plane Wave," Bull. Math. Biophys. 17: 

 35-40 (1955). 



b. "III. Relationship of Breakdown Curves and Mechanical Q," 

 Bull. Math. Biophys. 19: 1-7 (1957). 



3. Lombard, D. B., "Ultrasonic Rupture of Erythrocytes," Thesis, Penn- 

 sylvania State University (1955). 



4. Binstock, L., "Photographic Studies of Erythrocytes in Ultrasonic Fields," 

 Thesis, Pennsylvania State University (1960). 



