Large deformation of isotropic biological tissue

1. Bwmcchnmcs.

19’2. Vol. 5. pp. 6431-606.

LARGE

Pergamon Press.

PrintedinCireat

Britain

DEFORMATION OF ISOTROPIC BIOLOGICAL TISSUE* ROGER

W. SNYDER*

Engineering Science Program, University of Notre Dame, Notre Dame. Ind. 46556, U.S.A. Abstract-The requirements for the development of a general potential function describing the behavior of homogeneous, isotropic biological tissue are discussed and such a function is proposed. It is shown that this function yields both tensile and torsional results compatible with existing data.

INTRODUCTION

to the framework established to study large deformation (see, for example Green and Adkins, 1960), the response of material to various forces can be described in terms of a potential function. Once this function is determined, which is extremely difficult, a number of important problems can be solved. A number of attempts have been made to determine such a function for biological material. For example see articles by Blatz et al., 1969 Gou. 1970 Hildebrandt et al., 1969 and Simon et nl.. 1970. In general. however, these functions have been developed for a specific loading pattern. A more general function would permit the study of a number of types of loads and the interaction between combined loads. This paper represents an initial attempt at deriving such a function. A variety of techniques have been utilized for determining such potential functions. The function can be expressed as the sum of the dilatational energy (that required to produce a volume change) and the distortional energy. It can also be expanded into a series form. A discussion of these methods can be found in Eringen (1967) or Green and Adkins (1960). A semi-inverse method will be employed in this paper. That is,

ACCORDING

the form of the function for tensile loading will be determined. Then a more general form will be proposed and used to study torsional loading results. BASIC RELATlONSHIP

Tensile tests conducted on a wide range of soft biological tissues indicate a nearly straight line relationship between the slope of the force extensior curve and the force. In terms of the Lagrangian stress, T, (force per unit undeformed area): and the extension, A, (deformed length divided by the undeformed length); this can be written as (see Fung, 1968): dT x=

aT-l-k.

Integrating this equation yields, T

=

i

[@A-1)

_

l]

where the integration constant has been determined from the zero stress condition at A = 1. The resulting introduction of the - 1 factor into equation (2) gives rise to some difficulties in determining k and a from experimental data. The method of differential correctors,

“Received 1/Vovember 197 1. *This work was carried out in partial fulfillment of the requirements for the degree of PhD at the University of Notre Dame and was sponsored in part by National Science Foundation Grant No. BO 12 122 and General Electric RESD. Currently a Visiting Investigator with Howard Hughes Medical Institute at Peter Bent Brigham Hospital. 721 Huntington Ave., Boston. Mass. 02115. 601

602

ROGER

W. SNYDER

such as described by Pennington (1965) can be used to fit equation (2) to data points. One of the difficulties in calculating the parameters k and a is in determining the zero extension or initial length of the specimen. Equation (2) can be modified to account for a small initial load, such as is used to hold the specimen while measuring the length. If Li is the measured length and L,, is the actual undeformed length, then; Lb= x*L()

(3)

where, from equation (2); A* = lfiln

aT*

[

T-Cl

1

(4)

and T* is the initial Lagranian stress. The true extension is defined by; A= -= L

Lo

A*L -=

Lo

h*A’

the undeformed are related by;

area and the measured

A0 = A*A;.

area

(7)

Thus the actual Lagrangian stress is given by;

where T’ are the reported values of the Lagrangian stress, based on the pre-strained area. k and a can be determined by using the uncorrected data. Then A* can be computed and the data corrected according to equation (8). This process can be repeated until the desired accuracy is achieved. The effect on the results shown in Fig. 1 is slight. Once k and a have been determined, equation (2) describes the tensile stress extension relationship. The corresponding energy function can be found by integrating this relation-

where A’ is the measured extension. Substituting equation (4) and (5) into equation (2) yields; T

=k [taT*+

l)"'p'"'-"1

(6)

k and a can then be determined by the previously mentioned method of differential correctors. True stress extension curves, based on the same set of values of force and extension but corrected for different initial loads are shown in Fig. 1. The original data is taken from StenKnudsen (1953). Figs. 45 and 48. The results are from experiments on the frog’s anterior tibia1 muscle. This muscle was mounted in a bath of modified Ringer’s solution (maintained at 0°C and pH of 7.2-7-4) and subjected to simultaneous tension and torsion. There will be an additional effect on k and a if the crosssectional area is also measured in the prestrained state. If the tissue is incompressible,

Extensaon ,

A

Fig. 1. Effect of initial load on true stress extension curve (data from StewKnudsen (1953), based on tensile tests of frog’s anterior tibia1 muscle).

LARGE

DEFORMATION

OF ISOTROPIC

ship. Thus; (10)

BIOLOGICAL

TISSUE

603

where x is the axis in the direction of the force. The invariants are given by: I,_, = A,‘+ 2ht’

where /3 reduces to A under uniaxial deformation, In addition to the above requirement on /3, the energy function for a homogeneous isotropic incompressible material must be a single-valued function of two of the three invariants of any one of several deformation or strain tensors (see, for example Eringen, 1967). Choosing, for convenience, the inverse of Cauchy’s deformation tensor, a suitable function is; fi=&

[I,++

The deformation

(I;_,-311,_,)“?].

(11)

III,_, = A,%( = 1. where the condition of incompressibility been utilized in the final equation. Equation (13) yields: T,, = 2[Ac-&][z+;g]

has

(16)

where p has been evaluated from the condition of zero transverse stress. From equation (10);

tensor is given by;

a2 a?

ciil = --aXK

(12)

aXK

where xk is the coordinates of the deformed shape and XK the coordinates of the initial shape. The Lagrangian stress can be found in terms of the invariants of this tensor from Finger’s relationship (see Eringen, 1967):

TEim = -p&m

(1%

+ 2q,,, aw_2c. 1

aI,_,

aw

X

I]

(17)

Finally, equations (11) and (15) yield; p=A

+-$L c

’

-1

(18)

1 a0 w -=-aII,_, 2 al,_,

axK

AmaIr,_, 1 -ig (13)

[ea(8-*)-

If equations ( 18) are substituted into equation ( 17); then the result is equation (2). Thus, not

where p is introduced to compensate for the indeterminate relationship between hydrostatic pressure and volume change for the incompressible material.

only does the function proposed in equations (10) and (11) yield the correct energy function under uniaxial tension but also the correct stress.

Uniaxial tension The deformation described by;

Simultaneous in simple tension can be

x= h,X v= A,Y ; = A,2

extension and torsion

Simultaneous extension and torsion cylindrical rod can be described by:

of a

r=R/fi

(14)

8=0+AcvZ 2=AZ

(19)

ROGER W. SNYDER

604

where h is the extension and LY is the twist per unit length as measured in the extended state. The invariants are given by; I,_, = ; + $h’& + x” II,_, = 2h+$+r”W

(20)

III,_, = 1. Again p can be determined by the condition that T,., is zero on the transverse surface. For the complete solution, see Green and Adkins (1960). If the resulting surface tractions are integrated over the end surface, the following force and moment system results:

a0 r dr +i [ aII,_, 11 1

f2 [&]]

r= 2xXry2

r3dr}[ea@-“-

+ [-$--]]

r3 dr} [e”P-“-

(21)

I]

l]

(22)

w

3 I,+

(23)

alI,_,

-

1,-,~1~-~-311,-,~

P = -Ir,_,+211,-,(I:_,

0.0163 cm.

(24)

If the initial torsional stiffness at the point of indifference (0.02 dyn-cm) is used, then;

where r. is the deformed value of the outer radius. The derivatives of p are given by: -+“+ C1

response of the muscfe. Since both pure extension and torsional results were reported, the tensile data can be used to determine k and a. An additional reason for selecting this particular set of data is that it shows a straight line relationship between torque and angle of deformation. This is in apparent contradiction to other data (for example, Fung er al., 1966) which indicate an exponential relationship similar to that found for extension. All values of torque, load, torsional stiffness, etc. were normalized with respect to their values at the point of indifference. This is defined as the extension at which the extra tension during isometric contraction is zero and is highly reproducible. Based on the reported value of the Lagrangian stress at this point (2.94 X 106dyn/cm2), it was determined that the specimen radius was;

3 -3II,_,)

-

Note that according to the last term of equation (21), there is an end force from the angular deformation. This is known as the Poynting effect. Sten-Knudsen ( 1953) has conducted extensive tests on the torsional

r=

0*0181cm

(2%

appears to be the specimen radius. Both values were used in subsequent calculations. Figure 2 shows the initial torsional stiffness, found by dividing a small torque increment by the corresponding angular deformation, vs. the extension. The agreement is good for both values of the radius, although it is obviously better for the second value. Figure 3 is a plot of the torque vs. the angle of twist (the torsional stiffness being the slope of this curve). The results do indeed show the straight line relationship indicated by Sten-Knudsen (1953). In fact, this relationship holds for at least four times the angle of twist shown in Fig. 3. Figure 4 indicates the effect of increasing the specimen radius by a factor of ten. At this point the exponential nature of the material becomes apparent, but only at large angles of twist. Thus it is the specimen size which causes

LARGE

DEFORMATION

OF ISOTROPIC

BIOLOGICAL

605

TISSUE

zo-

I a--

r =0.181 cm kx45-9 9 ’ 5 99

04-

OZ-

Extension , X Fig. 2. Initial torsional stiffness of frog’s anterior tibia1 muscle (data points shown from Sten-Knudsen, 1953). 40

8 II ! 0

0

?

4,

so

-r

rc

Total

angle

r

*o-0181 cm

x

=4590.0

of rwist

,

radians

Fig. 3. Torque versus angle of twist for a series of values of extension (see equation 22).

Total angle

Fig. 4. Effect

of twst

,

rOdlOflS

of increased specimen size on torque versus angle of twist curves.

the results of Sten-Knudsen (1953) to differ from other similar experiments. CONCLLMONS

It can be seen from Fig. 2 that the isotropic, incompressible model, as given by equations (10) and (11) is adequate to describe the response of this muscle fiber to torsion. Although muscle fiber is basically transversely isotropic, it would appear that, at least for this muscle fiber, the anisotropy has little effect on the torsional stiffness. It should be noted that the deformations studied were isochoric (zero volume change). Thus these results would not reflect any compressibility. Some recent data by Fields (1970) suggests that biological tissue may not be incompressible. These deformations would not indicate this factor. Experimentally, simultaneous transverse measurements may be required to settle this question. Analytically, the third invariant of the deformation tensor must be included in j3. In addition a slightly different form of Finger’s

606

ROGER

W. SNYDER

relationship must be used in solving the problems discussed above. These changes are currently being investigated and will be reported in a subsequent paper.

REFERENCES Blatz, P. J., Chu, B. M. and Wayland. H. (1967) On the mechanical behavior of elastic tissue. Trans. Sot. Rheof. 13,83- 102. Eringen, A. C. (1967) Mechanics of Conrinrm. John Wiley &Sons, New York. Fields, R. W. (1970) Mechanical Properties of the Frog Sacrolemma. Biophys. J. 10,462-479. Fung, Y. C. (1968) Biomechanics: Its scope, history and some problems of continuum mechanics in physiology. Appl. Mech. Rea. 21, No. 1. Fung, Y. C., Zweifach, B. W. and Intaglietta, M. (1966)

Elastic environment of the capillary bed. Circ. Res. 19, 441-461. Gou, P. (1970) Strain energy function for biological tissues. J. Biomech. 3,547-550. Green, A. E. and Adkins. J. E. (1960) Large Elastic Deformation and Non-Linear Continuum Mechanics. Oxford Universities Press, London. Hildebrandt, J., Fukaya, H. and Martin, C. J. (1969) Simple uniaxial and uniform biaxial deformation of nearly isotropic incompressible tissues. Biophys. J. 9, 781-791. Pennington, R. H. (1965) Introductory Computer Methods and Numerical Analysis. The MacMillan Co., New York. Simon, B. R., Kobayashi, A. S., Strandness, D. E. and Wiederhielm, 0. A. (1970) Large deformation analysis of the arterial cross section. ASME Winter Annual Meeting. Sten-Knudsen. 0. (1953) Torsional Elasticitv of the Isolated Cross-Striated Muscle Fibre. Acta Ph)siol. Stand. 28, Suppl. 104.