as the properties of the biological tissue has been found analytically and in explicit form. Equation (22) is sufficiently simple and convenient for practical engineering calculations of the ideal cutting parameters. It establishes the relationship of the mean cutting velocity to the following parameters: I) force of the surgeon F x applied to the ultrasonic instrument; 2) mechanical properties of the biological tissue which are determined by its force of resistance to cutting fo; 3) amplitude A and frequency ~ of ultrasonic oscillations of the cutting block of the instrument; and 4) inertial property of the instrument and the acoustical unit, which is determined by which part of their mass oscillates. The theory developed and the model constructed will find use in the development of the optimal technology for ultrasonic surgical incursion in soft biological tissues and of the ultrasonic instruments required for t~is purpose. The authors express their gratitude to G. A. Nikolaev for assistance in posing the problem and discussing the results of this work. LITERATURE CITED i.

2.

3. 4.

V . A . Polyakov, G. A. Nikolaev, M. V. Volkov, V. I. Loshchilov, and V. I. Petrov, The Ultrasonic Welding of Bones and the Cutting of Living Biological Tissues [in Russian], Moscow (1973). V . I . Loshchilov, S. M. Volkov, V. V. Zasypkin, and V. P. Borisov, "The ultrasonic cutting of biological tissues," in: Ultrasound in Surgery [in Russian], Moscow (1973), pp. 24-29. V . I . Loshchilov and S. M. Volkov, "The mechanism of the ultrasonic cutting of biological tissues," in: Ultrasound in Surgery [in Russian], Moscow (1973), pp. 29-33. V . P . Borisov, "The development and study of the ultrasonic cutting of soft biological tissues," Author's Abstract of Candidate's Dissertation in the Technical Science, Moscow (1975).

DEFORMATION OF THE HUMAN SKULL UPON IMPACT (AN EXPERIMENTAL STUDY AND SOME PROBLEMS IN MODEL DESIGN).* 2.t

THE DYNAMICS OF THE SKULL BASE UPON IMPACT LOAD OF THE VAULT A. S. Barer, Yu. G. Konakhevich, L. N. Sholpo, V. Kh. Petlyuk, and N. A. Uglanova

UDC 611.71:620.1

In the model of the human skull as an ellipsoid shell with a flat base [2, 3], special interest clearly lies in the dynamics of the most yielding segment of this structure, namely, the central zone of the base. Considering this feature and also the proposed special role of the skull base in the pathogenesis of craniocerebral trauma discussed in previous work [4, 5], recording of ~he accelerations at the central point of the skull base upon impact load of the vault was carried out in all the experiments. From the viewpoint of the proposed action on the basal structures of the brain, greatest interest lies in the relative movement of the central point of the base in a coordinate system of the skull vault, i.e., the "true" deformation of the skull in the direction perpendicular to the plane of the base [3, 5]. Upon impact to the forehead and the occiput regions, the first local extremum in the acceleration of the central point of base, as a rule, is reached at the instant of maximal acceleration in the skull vault in the counterimpact zone. For the most vigorous impact (duration of the impact pulse in the vault of about 2.5"10 -3 sec and less), the local ex*Presented to the Second All-Union Conference on Biomechanics, Riga, April 1979. *For Part 1 see [i]. SThe experimental method, scope, and recording scheme were discussed in our previous work [i]. Moscow. Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 525-529, MayJune, 1980. Original article submitted May 28, 1979.

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tremum in the base may lag somewhat relative to the maximum in the vault. For the softest impact (duration of about 7"10 -3 sec and more), tile extremum may be somewhat in advance of the maximum in the vault. It is interesting to note that upon collision by the forehead region with a barrier, the first local extremum is always positive and is an absolute maximum of the acceleration of the central point of the base relative to the vault. (For acceleration in the skull vault, positive is taken as the direction from the apparatus and for acceleration in the base, as the direction towards the exterior from the skull cavity.) Upon collision by the occiput region with a barrier, the first local extremum is always negative and is an absolute minimum in the acceleration. For impact to the temple region, the first local extremum in the acceleration in the base is always positive but is several times less in absolute magnitude relative to the second local extremum, which is always negative and is'the absolute minimum in the acceleration of the central point of the skull base. The extremal displacement of the central point of the skull base relative to the vault always coincides in direction with the absolute acceleration extremum (i.e., is positive for the forehead region and negative for the occiput and temple region impact sites) and is reached at times from 4.5"10 -~ to 9.0"10 -3 sec from the instant of the onset of collision. Special importance lies in the proper selection of an experimental parameter which sufficiently reflects the features of the dynamics of the skull base and correlates well with values which characterize the impact load of the vault. Two carefully studied experimental terms, namely, the acceleration and displacement of the central point of the base in the skull vault coordinate system, should be considered for this parameter. It should be noted that the deformation is a more obvious choice as an index for the proposed action on the basal structures of the brain than the acceleration of the central point of the base, which reflects this action more indirectly. Nevertheless, in this stage of the study, the use of deformations as the basic parameter produces a series of serious objections. While the acceleration of the central point of the base is a directly recorded parameter with relatively low experimental error, the deformation of the base at this stage may be determined only by double numerical integration of the acceleration oscillogram and double interpolation is required due to the limited resolution of the measuring tract. The calculation error in this case rises unjustifiably. It is also significant that while rather simple, reliable, and close-to-linear correlations may be found for the extremum values of the acceleration on the parameters of the impact load of the vault, such correlations for deformations, though reliable, are significantly nonlinear, which complicates the results obtained and lowers their accuracy. Thus, in this stage of the work, it was thought desirable to limit ourselves to use of the extremal values of the acceleration of the central point of the base as the major parameter which describes the result of impact load of the vault. We note that such a limitation does not apparently lead to a significant distortion in the results obtained since a reliable, though significantly nonlinear, correlation exists between the extremal values of the acceleration and the deformation of the central point of the base. The correlation terms relating the experimental values of the absolute extrema of the acceleration in the skull base and the collision velocity are given in Table i. These correlations were significant and close to linear for the impact sites and collision surfaces studied. Thus, we may assume that the extremum value of the acceleration is reliably found with some statistical scattering by the ordinary type of linear regression equations, whose free terms bo and proportionality terms bl are also given in Table 1 for each impact site and collision surface. We immediately note that a significant, close-to-linear correlation exists between the proportionality coefficients in the linear regression equations and the rigidity of the collision characterized by the parameter n. This finding permits us to approximate the experimental data for impacts of various rigidity by a single equation of the type:

a,~xt=bo+bln + b2vo+batzvo,

(1)

where aex t is the maximal acceleration in the skull base, m/sec2; and Vo is the collision velocity, m/sec. The coefficients in Eq. (i) are found by multiple regression analysis if we adopt the term n as the third variable. The values for the coefficients of Eq. (i) are presented in Table 2.

383

TABLE i. Regression Curve Parameters Relating the Acceleration at the Skull Base and the Collision Velocities for Various Types of Collision Surfaces Impact site

Collision surface material

CoTrelation Free coeffi- I term !c i e n t ! . .

Meansquare

lerror . .

Propor - Mean tionality square ccient oeffi-- I error . . I

|

Forehead

Occiput

Temple

Note.

Wood Brick Steel Wood +felt I Wood + felt II Brick + anozoz Steel + felt I Steel Wood + felt I Wood + felt II Steel + felt I Wood brick Steel

0,963 0,832 0,992 0,998 0,996 0,998 0,987 --0,985 --0,690 --0,984 --0,974 --0,831 --0,938

I-0,948

I -299,3 I --266,3 [ 3,6 I --295.9 --481,2 [ --432,9 I --246,1 148,0 --181,2 405,8 295,4 10,7 47,6

I I I I

99,5 I

140,0 703,1 196,0 655,8 44,2 1008,0 15,6 566,2 22,1 482,3 I 26,2 892,5 I 571,2 I 49,0 76,7 --931,1 I 210,3 --717,6 I 58,7 --556,7 / 64,1 --722,3 / 20,8 I --108,0 I 16,4 I --176,4 I !

16,7 1-248,8 I

315,6 408,9 163.0 28,0 34,0 51,5 103,2 204,9 714,4 97,4 112,1 36,4 47,6

43,3

Felt I was i0 mm thick and felt II was 20 mm thick. TABLE 2. Multiple Regression Parameters Relating the Acceleration at the Skull Base and Collision Velocities Impact site Forehead Occiput Temple

b:

721,5 --946,3 413,1

--1899,9 2570,9 --671,3

-

1151,3 1869.4 --723,6

-- 1205.7 2646,7 996.3

Equation (i) with the coefficients presented in Table 2 satisfactorily describe the experimental dependences of the extremum acceleration of the central point of the skull base on the collision velocity and collision rigidity characterized by the parameter n. It is clear, however, that both from the viewpoint of accuracy and understanding of the essence of the phenomenon studied, discovery of a functional relationship of the parameters examined by means of a mathematical model is preferred. Such a model should, of course, satisfactorily describe the experimental data in a broad range of possible collision conditions, qualitatively correspond to our concepts of the biomechanics of the phenomenon studied, and, finally, should be maximally simple and convenient for practical use. In the literature available to us, we have virtually not encountered attempts to produce a mathematical model of the cranial biomechanics upon impact load considering the special role of the skull base. An exception is found in the studies carried out in the previous stages of the present work [i, 2], in which an attempt was made to represent the deformation of the skull base by a model with distribution parameters. However, the corresponding equations were rather cumbersome and their identification, taking account of the available experimental data, is very difficult. A model of the joint system of the skull base [6] is also extremely interesting but its practical use for testing protective headgear involves great difficulties. On the other hand, analysis transfer coefficients of the vault--base system determined by the pulse excitation method indicates the possibility of a satisfactory representation of the dynamics of the skull base by a relatively simple two- or three-element viscoelastic linear model with concentrated parameters. The limited resolution of the measuring tract and the statistical scattering of the experimental data which is generally characteristic for biomechanical spatial systems leads to a marked indefiniteness in determining the amplitudes (but significantly less ambiguity in determining the frequencies) of the major resonances of the system. Thus, the direct use of standard methods for the identification of a model with concentrated parameters according to logarithmic transition characteristics may lead to significant errors in the determination of the coefficients of the model equation.

384

Nevertheless, the data obtained permit reliable establishment of at least two resonances at frequencies 300 • i00 and 800 • 150 Hz in the frequency range studied from 50 to 2500 Hz. We note that the higher of these frequencies apparently corresponds to the major frequency of the natural oscillations of the human skull, which is in good accord with the experimental data obtained in the vibroloading of a head mannequin [7]. It is significant that in our experiments on the impact loading of an empty, isolated skull, the corresponding frequency is significantly higher and is about 1050 • 150 Hz, which also corresponds to the literature data for the head of a biomannequin with removed brain matter. The calculated frequency of the fundamental tone of the previously proposed skull model in the form of a dome with a flat base is about Ii00 and 850 Hz for the empty and filled shell, respectively [2], which is in good agreement with the experimental data of the present stage of the work. (The corresponding physical model for testing protective headgear is presently in the stage of development and testing.) More complex behavior is found for the third resonance frequency found in all the experiments for the temple site and is 1200 • 200 Hz for the filled skull. A significant resonance is found in this region of the experimental curves obtained for impact at the forehead and occiput sites only in about one-half the cases. While the bands at the two lower frequencies are easily found even upon a qualitative examination of the oscillograms for the acceleration in the base in the aftereffect period (when the oscillations in the vault become negligibly small), this is not possible for the third frequency. The causes for this effect are not yet clear and, although we might expect that the effect of the natural vibrations with half-life an order of magnitude less than the duration of the exciting impact pulse is unlikely to be significant, it is worthwhile to provide for the possibility of using the three-element model of the phenomenon studied. A number of effects related to the transformation of the impact pulse in the vault-base system may be found even for the most simplified, purely elastic two-element model. We may show that solutions of the equations of this model qualitatively provide a rather correct description of the nature of the dependence of the acceleration of the central point of the skull base on time for values of the major frequencies of the free vibrations of the model corresponding to the two lower resonance frequencies obtained in our experiments. Although the calculated extremal accelerations, not considering the effect of viscosity and the possible effect of a third element, naturally greatly exceed the real values, the distribution of the calculated extrema over time and the ratio of the amplitudes of consecutive extrema in the period of nonzero external action and in the near aftereffect satisfactorily correspond to the experimental data. Introducing the relative acceleration of the central point of the skull base in the coordinate system of the vault, we may show that for impacts to the forehead and occiput sites, the first extremum for this term (which, taking account of damping, is maximal in modulus) should virtually coincide in time with the maximum acceleration in the skull vault, which also is in accord with the experimental observations. The approximate value of the constant attenuation time for both resonance frequencies may be evaluated directly using the experimental oscillograms and is about (3.5-5.0)'10 -3 sec in both cases and, though the effect of the nonzero viscosity on the time distribution of the extrema for the acceleration in the base is apparently small, the discrepancy between the model and experimental acceleration amplitudes due to damping is extremely significant and increases with increasing time. We may expect that the correct selection of parameters for the equations of the threeelement viscoelastic model of the type:

~=

\'2 (Y2-YJ)

+k2(y~--y,) +v,

(~--9~) +k~ ( y - y , ) "

(2)

n~92 = ~'2 (0~ - 02) + k2 (gt - ~2) + v3 ( 0 3 - 0~) + k3 ( g 3 - y2) ;

(3)

~393=u (02--03) +k3 (~/2--~3) ,

(4)

where ~ i i s t h e mass, kg; Yi, d i s p l a c e m e n t , m; k i , e l a s t i c i t y , N/m; and v i , v i s c o s i t y , kg/ s e e ; model e l e m e n t s ( i = 1, 2, 3) p e r m i t us to p r o v i d e a s a t i s f a c t o r y a p p r o x i m a t i o n of t h e e x p e r i m e n t a l d a t a f o r t h e r e a c t i o n of t h e s k u l l base to t h e impact l o a d i n g of t h e v a u l t . Equaeions ( 2 ) , ( 3 ) , and (4) were t e s t e d on t h e Dnepr MPT-9 computer u n i t and t h e p r e l i m i n a r y results of model "experiments" qualitatively support this assumption. Such a model may be used for preliminary calculated evaluations of protective headgear and apparently will permit some refinement in our understanding of the biomechanics of craniocerebral trauma.

385

The authors thank B. A. Rabinovich and V. I. Kharchenko for valuable discussions and also G. K. Derimarko for assistance in translating the model to an analog-digital computer program set. LITERATURE CITED i.

2.

.

4. 5. 6.

7.

A. S. Barer, Yu. G. Konakhevich, L. N. Sholpo, B. Kh. Petlyuk, and N. A. Uglanova, "Deformations of the human skull upon impact: An experimental study and some problems in model design, i. The technique for studying the biomechanics of the human skull upon impact," Mekh. Kompozitn. Mater., No. 2, 319-324 (1980). A~ S. Povitskii, B. A. Rabinovich, V. M. Tardov, V. A. Cherneikin, and L. N. Sholpo, "A flat-base dome as a model of the human skull upon impact," Biofizika, 19, 1087-1091 (1974). V. A. Cherneikin and L. N. Sholpo, "A further study of a model of the human skull as a dome with a flat base," Biofizika, 21, 376-381 (1976). B. A. Rabinovich, L. N. Sholpo, and E. Ya. Shcherbakova, "The dependence of the nature of craniocerebral trauma on impact conditions," Kosm. Biol. Med., No. 5, 62-66 (1971). Yu. G. Konakhevich and L. N. Sholpo, "The dependence of the state of consciousness in craniocerebral trauma on impact parameters," Vopr. Neirokhir., No. 2, 32-36 (1978). A. P. Gromov, O. S. Saltykova, G. S. Bolonkin, and N. P. Pyrlina, "The dependence of osseous skull deformation on impact conditions," in: Biomechanics [in Russian], Riga (1975), pp. 5-9. E. S. Gurdjian, V. R. Hodgson, and L. M. Tomas, "Studies on mechanical impedance of the human skull: Preliminary report," J. Biomech., 3, No. 3, 239-247 (1970).

CHANGE IN THE BIOMECHANICAL PROPERTIES IN THE BONE OF RATS AS A RESULT OF A 19-DAY SPACE FLIGHT IN THE KOSMOS-936 SATELLITE* G. P. Stupakov, A. I. Volozhin, V. V. Zasypkin, and S. M. Remizov

UDC 611.71:541.68

The importance of the effect of prolonged weightlessness on the bone system is a result of the possibility of a reduction in the strength characteristics of bones. The literature data on this problem are based on terrestial experiments in which some method is used to serve as a model for the effect of weightlessness on the skeleton [i, 2]. The changes in the properties of the skeleton by the action of weightlessness have not been studied adequately. The only established fact in this area is the negative balance of mineral salts and development of osteoporosis of the heel bone in humans as a result of prolonged space flight [3, 4]. Osteoporosis of the tubular bones was found in rats in an experiment on a biological satellite [5]. In a study of the autopsy material from the Salyut-i space station, no change was found in the mass ratio of mineral and organic components and in the microhardness of the bone tissue in the bones of cosmonauts [6]. Studies of the degree of bone porosity, bone structural strength characteristics, and the biophysical properties of the bone tissue of biological specimens after the action of prolonged weightlessness have not been carried out. The aim of the present work was a study of the direction of change of the biophysical and mechanical properties of the weight-bearing skeleton of rats exposed to the effects of space flight in the Kosmos-936 satellite. The experiments were carried out for SPF rats with initial mass 206 • 7 g. A "synchronous" experiment was begun four days after launching of the satellite under terrestrial conditions by maintaining the animals in a satellite model in which conditions similar to the environment during space flight were imitated. The control animals were maintained in a vivarium. Bone matrix from other rats of the same line was implanted at 18-23 days after *Presented at the Second All-Union Conference on Biomechanics, Riga, April 1979. Moscow. Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 530-537, MayJune, 1980. Original article submitted May 7, 1979.

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