MECHANICAL BEHAVIOR OF THE HUMAN ANNULUS FIBROSUS* HAN-CHIN Wut

and REN-FENG YAO~

Division of Materials Engineering, The University of Iowa. Iowa City. Iowa 52242, U.S.A. Abstract-The mechanical properties of the human lumbar annulus fibrosus is investigated both theoretically and experimentally. The theory presented is based on the assumption of material incompressibility which is then justified by the experiments. Spencer’s theoretical approach is followed. but the fibers of the material are considered to be extensible. A strain-energy function and the associated constitutive equations are also presented. An optical method is employed _ . to measure the extension ratios and the change in relative orientation of the &ers.

1. INTRODUCTION

the main approach proposed by Adkins (1956), Green and Adkins (1960), Rivlin (1964) and Spencer (1972) for fiber reinforced material. The material is assumed to be incompressible and nonlinearly elastic. The fibers are considered to be extensible in the present investigation. and a strain-energy function is also presented. The purpose of the experimental study is to provide the material characteristics which appear in the theoretical analysis and to check the assumption made in the analysis regarding material incompressibility. An experimental stress-strain curve for the annulus fibrosus is also presented in this paper.

From indirect evidence, intervertebral discs are suspected as the site or origin of low back pain. In order

to understand the mechanical response of the intervertebral disc, material properties of each component of the disc must be determined. The annulus fibrosus is the outer fibrosus part of the intervertebral disc. Horton (1958), using a polarizing microscope and a comparative high intensity Xray beam, revealed that the annulus fibrosus is a series of concentric encircling laminate and that the collagen fibers run a uniform course in each laminate. The fibers of adjoining laminate cross each other at an angle of approximately 60”. The space between the collagen fiber framework of the annulus fibrosus is filled with ground substance, which in turn traps large amounts of water within and is an amorphous, mucopoly-saccharide protein complex (Galante, 1967; Harkness, 1966). In response to external forces, the annulus Bbrosus is a major component of the intervertebral disc. In this paper, the tensile property of the lumbar annulus fibrosus is investigated both theoretically and experimentally. A constitutive equation is presented for fibrous materials consisting of two families of fibers under finite elastic deformation. The mechanical response of the annulus fibrosus to external loading is complicated by its laminated and fibrous structure and by its timedependent property. It is shown, however, that the present theory does provide reasonable agreement with the experimental elastic response of the annulus fibrosus. The theoretical development of this paper follows

2 EXPERIMENTAL EQUIPMENT AND PROCEDURE

Human postmortem annulus fibrosus specimens were tested in this investigation. Specimens were selected from normal discs of subjects whose ages varied from 20 to 30yr old. Rectangular blocks were cut from the posterior or anterior annulus fibrosus of a lumbar intervertebral disc. Each block had the length of 31 mm, the width of the height of the disc and the thickness of the annulus fibrosus. These blocks were then embedded in gelatine (Table 1B and frozen at - 30°C for 12 hr. The blocks of gelatine were then sliced into several thin slices using a meat slicer (Hobart(j). A displacement gage was attached to the slicer. Specimens of identical thickness could therefore readily be produced. In order to obtain specimens that contained approximately equal amounts of fiber from both families, the thickness of the specimens was chosen to be 2mm. A dial caliper (Starrett), with an accuracy of

’ Received 10 March 1975.

Table I. Embedding gelatine solution

t Assistant hofessor.

f Graduate Assistant. 8 The writers are thankful to Dr. Earl F. Rose, Department of Pathology, University of Iowa, for providing the formula and the assistance in the g&tine embedment, and for granting the privilege of using his meat slicer and for his help in the slicing of the specimens.

B.M. 9/1-A

Cellosolve (ethylene glycol monoethyl ether) Capryl alcohol 1% Thiomersal (antiseptic) Gelatine (8@-1tHlbloom) Water

I

240 ml 30ml 20 ml lfJ@Jg 5800 ml

HAN-CHIN WV and REN-FENGYAK

2 ,r FIBER ORIENTATIONS

i

31mm

_I

taken under a microscope (Bausch & Lomb 904(YlD). The third extension ratio in thickness was measured by a dial caliper (Starrett). The experiments were conducted in a room where temperature varied from 68 to 75°F with a relative humidity of approximately 65%. 3. A THEORY OF TWO-FAMILY FIBROUS MATERIALS

Fig. 1. Shape of the cutting die. +0025 mm, was used to measure the thickness of the sliced specimen. Those specimens that were not cut parallel to the layers of the annulus fibrosus were discarded. A total of 30 specimens were selected for testing from the annulus fibrosus of fifteen lumbar discs. A modified press with a cutting die was designed and constructed to cut the specimens. The die was dumbbell shaped (Fig. I) with a reduced central area of 15 x 5 mm and expanded end segments of 6 x 7mm. The specimens were stamped to shape with the die, in the frozen state, ensuring the reproducibility of the dimensions. The fiber orientations are also shown in Fig. 1. A universal testing machine (Instron Model TM200), with an accuracy of &O-l% in the crosshead speed and ItO,S% in the load indicator, was employed to test the specimens. During loading, the crosshead speed was kept at 005 in/min. To minimize the effect of stress relaxation, readings were taken after the machine had been stopped and one minute had elapsed. The period of air exposure of the specimen was therefore about 10min which is within the time limit of negligible air exposure effect (see Galante, 1967). Special grips were designed for the machine to hold the small specimens without causing damage to the fibers. The specimen holders were made of plastic without serrations, and were disposable. Eastman 910 adhesive was applied to glue the specimen to the holders. The specimens were ready for testing 3 min after the application of adhesive. In order to examine the homogeneity of the deformation, two parallel lines were stamped on the surface of the central area of the specimen. An optical method was employed to obtain the strain (extension ratios) measurements. A camera (Nikon F) with extension tube was set up to take slides at each elongational increment (Fig. 2). The longitudinal and the transverse extension ratios were obtained by enlarging the pictures using a slide projector. The change in relative orientation of fibers when subjected to different loads were also measured using the same technique. These slides (Fig. 3) were * For the case of large extension ratios, unloading behavior is different from that of the loading one for the material considered. Thus, the theory of hypcrelasticity does not apply under this condition. The unloading hehavior of the soft biological tissues will be discussed at length in a subsequent paper.

Introduction The stress response of soft tissues has been the subject of many theoretical investigations. Under normal conditions, soft tissue such as the annulus fibrosus. skin and tendon material behaves in a nonlinear elastic fashion. Although stress relaxation has been observed, elastic response can be obtained after a long relaxation time, according to Fung (1967) and Blatz et al. (1969). Since the deformation of soft tissue is generally large, a finite deformation theory must be applied. The theory of hyperelasticity, appears to be applicable as a first approximation. (See for example, the work by Mooney, 1940; Adkins, 1956; Green and Adkins, 1960; Saunders, 1964; Hart-Smith and Crisp, 1967; and Rivlin, 1960, 1970.) In this theory, the material is assumed to be isotropic and homogeneous and the strain-energy function to be a function of the principal invariants. Valanis and Landel (1967) proposed a more useful expression for the strainenergy function using the extension ratios. The isotropic finite deformation theory has been applied to biological tissue by Fung (1967), Lee ef al. (1967), Crisp (1968). Hildebrandt et al. (1968), Blatz et al. (1969) and Veronda and Westmann (1970). In the case of the annulus fibrosus, the effect of fiber orientation must, however, be taken into consideration. Finite deformation theories of incompressible elastic material reinforced by inextensible cords have been developed by Adkins and Rivlin (1955), Green and Adkins (1960) and Kydoniefs (1970). In these theories, the cord is assumed to be so thin as to occupy no volume, and to be embedded in an isotropic matrix material. Since the fibers in the annulus fibrosus are densely distributed and the matrix (the ground substance) is very weak, the above theories are not applicable to the description of the mechanical behavior of the annulus fibrosus. Fortunately, a theory has recently been advanced by Spencer (1972) that is suitable for the present purpose. Following Spencer, the strain-energy function can be regarded as a function of the deformation gradients, &+/~X,., and the two initial fiber directions, A, and E, for the two-family fiber-reinforced material. Here, X, and xi are respectively the material and the spatial coordinates, and A, and E, are components of unit vectors A and B of the two families of fibers at time t = 0. If the two families of fibers are indistinguishable except for their directions, and fiber densities of the two families are the same, then by the theory of algebraic invariants, the strainenergy

Fig. 2. Simple tension test: (a) A, = 1.0, (b) AI = 1.2. Fig. 3. Fiber orientation: (a) 2@ = 57”, (b) 2P = 43”.

(Facing p. 2)

Mechanical behavior of human annulus fibrosus function W can be written as w = W(J,,Jz.J3

+ J&JS.Jfj.J,.J8.4.

(1)

where .I1 = trG, 52 = +((trG)’ - trG*), J3 = tr(AA)G*. J4 = tr(BB)G’, J5 = tr(AA)(BB)G, and

G,, =

SX,

iix

?Xi cz

r

IS

J6 = tr(AA)(BB)G’, 5, = tr(AA)G, JB = rr(BB)G, K = detG,

Fig. 4. Initial state of two-family fibers.

(2)

the right Cauchy-Green

Combining equations (7) and (8) one obtains 4 = isin- ’ {$%in2@}.

strain

(9)

\

Marerial with two-family extensible fibers The annulus fibrosus, will now be considered as a block of incompressible material which is reinforced by two families of straight, parallel, eitensible fibers and in' which the fibers are densely distributed. The effect of the matrix material will be neglected in the analysis. The fibers of the two families are initially inclined at an angle 2@ (Fig. 4) to each other. This angle is then denoted by 24 during deformation. The X 1 and X 1 axes are chosen to be in the plane of the two families of fibers and bisect the angles between the fibers of different families. The X3 axis is normal to the X,-X2 plane (Fig. 4). It is assumed that the plane of the fibers remains plane during deformation. The two unit vectors defining the initial fiber orientations are given in the present case by

A = (cos @,sin C 0), (3)

B = (cos @, -sin @,0). For simple tension, the deformation s1 = E,,X,,

-Yz= &X2,

The fiber orientation is thus specified by the above equation. In order to calculate the extension ratios. the reference state must be defined. Specimens of the annulus fibrosus are so soft that it is difficult to specify the reference state precisely. In this investigation. the reference state is defined as that state when the specimen begins to transmit load.* From the measurements, the current extension ratios i,,. E.?and R, are determined with respect to the applied load. The true stress (Cauchy stress) has been used in this investigation, and the following relation is observed:

(4)

where i.,, i.,, E.3are extension ratios. Since fibers are extensible, it follows that i.’ = i.: cos* @ + 1: sin* @,

(5)

where 1. is the extension ratio in the direction of the fiber. Since the deformation is homogeneous, the fibers in either family remain parallel and straight after a deformation. Let a and b be unit vectors of the two families of fibers at time t, then a = (cos 4, sin 4.0) and

(6) b = (cos 4. -sin 4.0).

F1

A

Ao 12~~

.F.

--y- = A, -

= n,fJo.

Ao

V . 1 . v, = A,A*A3.

(11)

and has also been calculated. Figure 5 indicates that the assumption of material incompressibility is reasonable for the annulus fibrosus. Figure 6 shows the change of fiber orientation during deformation. It is seen that the theory agrees well with experiment. In terms of the left Cauchy-Green strain tensor 9ij

axia.yj = ax, E,

(12)

*

the invariants in equation (2) reduce to J, = bg, J2 = i((trg)* - trg’), J3 = 1’ tr(aa)g, J4 = 1* tr(bb)g,

J, J6 5, K

= = = =

1’ cos 24, tr(ab), 1’ cos 2@ tr(ab)g, JB = I’,

e

12

detg

(13)

where cos# = *~cosO

and

sin4

1,

= IIsirt@.

(7)

From the assumption of incompressibility 1,121,

= 1.

(8) 01 0

* The deformation of soft tissues involves both fiber slipping motion in the matrix and fiber elongation. The initial fiber slipping motion cannot transmit load.

(10)

Here, F is the applied load. A,, and A are respectively the initial and current cross-sectional areas of the specimen. and u. is the nominal stress. The ratio of current volume, V, to the initial volume. V,,. is given by:

is described by -Ys= 13x,,

F

0= - = -

4

STRESS (r,.

psi x 10’

Fig. 5. Plot of &&,A3 vs stress u,.

HAN-CHINWV and REN-FENGYAO -

+= f sin-’

AlA2

( F

sin 20

)

(14), it is found that

i

R, = &(A: ws2 @ + 1’: sin’ @ - 1)

(0=30’) m 30 d

and

*- 20 w d f 10

(19) R2 =

t--

cos* @ - 1: sin2 @ - ws 2@X

and at the reference state

OO--' STRESS o,,

R, = R2 = 0.

psi x 10’

(20)

Equation (17) then becomes Fig 6. Angle 4 vs stress 6,.

W = W(R,, R2).

which reduce further to the following, for simple . tension, Jl = 1: + n: + n;2n;2, J2 = 2;’ + 12;’ + Jilt;, 53 = J4 = 1: cos2 Qi + A: sin’ @, JS = cos 2@($ coo52@ - II: sin2 @,x J6 = cm 2@(# cos’ @ - A; sin’ @),

(14)

(21)

It is well-known that the constitutive equation for a hyperelastic material is given by:

For the constraint of incompressibility, a Lagrangian multiplier P is introduced such that W is replaced by W + fP(K - 1). Thus,

J,=Js=~~cos2@+~$sin2@, K = 1, where K = 1 is the constraint due to material incompressibility. The following equations may be obtained from equations (14) 2 cos 2@ ws2

cos 29

@‘nf,

(15)

(23) Using equations (7) (13) and (19X equation (23) takes the form -g+ 1

61 =gws2@

- J6 = 2 ws 24, sin2 @A;.

ws2d&-

aw

aw

Thus, n: =

wsZQ(J3

a2 = 32; sin’ @ -ws2!DdR, aR,

+ J4) + 2Jtj

+P,

(24)

a3 = P, (9

A2 _ 2-

+ P, 32

ws 2@(J3 + J,) - 2J6

In this way, J1, J2, Jg, J, and J3 can be expressed in terms of J3 + J4, J6 and cos2 R Since the initial angle 2@ and therefore the value of cos’ @ are known for a given material, the strain-energy function W is, using equations (1) and (14), given by:

where 6, = crii. ~7~= az2 and b3 = us3 are the principal stresses. In equation (24), a3 = P, which is a consequence of the assumption that the plane of the fibers remains plane during &formation. Using equation (19) equations (24) become now o,=U,$+P,

1

(25)

a2=2A2;+P, W = W(J3 + J4, J6).

(17)

It is now convenient to introduce two new invariants R, and R2, such that:

Rl=;(“+-l) and

(18) R2 = + (J6 - ws2 2@),

where 5 is a constant scale factor which makes R, and R2 small. In the present case, r = 10. Using invariants

2

a3 = P, 61 = $2: ws2 @[R,(2k,

+ k, ws 249.

where P is an arbitrary hydrostatic pressure. It is interesting to note that the form of equations (25) is the same as that derived by Valanis and La&e1 (1967) for isotropic hyperelastic material. Strain-energy jimction The strain-energy function is detined by equation (21). To characterize elastic soft tissues in a state of

Mechanical behavior of human annulus

5

fibrosus

In the case of a simple tension uz = u3 = 0, equations (28) render

;;

R, =

‘,C

test, u1 + 0

and k5 ws 2@ - 2k,

k, - Zk, cos 29 0.1

OF O

)I

(30)

0.2

RI=~(A: cos’o-A:

(29)

”

si$~-cos’~)

Fig. 7. Relationship between invariants R, and R2.

Figure 7 shows a linear relationship between R, and R, from experimental data. Using equations (19). equation (29) becomes now 4A:k, cos4 9 + k, cos 2@

large deformation, the strain-energy function has been represented in two different forms: the exponential form (see, e.g. Fung (1%7), and the power form (e.g. Blatz et al., 1969). In the following, the power form has been chosen for convenience. In a Taylor series expansion of the strain-energy function, where R, and R, are small, the higher order terms can be neglected. Thus, W(R,,R,)

= k, + k,R, + k,R1 + k3R: + k& + k,R,R,,

- 2k,( 1 + cos 29 + cos2 2q [k, ws 24, - k,(l + cos’ 2@)]

(31) by substituting equations (19) and (31) into equation (30), a uniaxial stress-strain relation is then obtained as u, = c;i$1l: - l),

(26)

k,, = w(O,O);

k, = aw(O,O)/aR,;

k, = aw(O,O)/aR,;

k3 = a2 w(0, o)/aR: ;

k, = a2 w(0, o)/aR: ;

k, = a2 w(o,O)/aR,aR,.

k; - 4k;

c = ~ws40wsz29

[k, ws2@ - k,(l + ws2 2*)-J’

For small deformation, equation (32) reduces to 61 = 4CE, = EIE,.

In the reference state W(O,O) = 0 and o1 = u2 = CJ~= 0, and therefore k,, = k, = k2 = 0. Since the function W(R,,R,) is form-invariant in R, and R2, its expanded form must be symmetric and R, and R, such that k3 = k,. Thus, equation (26) reduces to = k,(R: + R:) + kSR,R2.

(27)

and equations (24) become

(34) where E, = ~ws4#ws22@

k$--4k; [k, ws2@ - k,( 1 + ws2 2@)]’

(35)

u2 = 41; sin’ @[RI& - k5 cos 2@) + Rz(k, - 2ks ~0s 2@)] + P, Ul = P.

(28)

E,=600

psi

The linear relationship between u1 and # - 1) of the human lumbar annulus fibrosus is shown in Fig. 8. Figures 7 and 8 in conjunction with equations (29) and (35) determine the coefficients in the expansion of the strain-energy function. They are found to be k3 = -570 psi and k, = 7980 psi. Thus, the strain-energy function of human lumbar annulus fibrosus is given by: W = 7980R,R,

/ 250

500

STRESS al,

7s

psi

Fig. 8. Stress crl vs ,I:(# - 1).

(33)

where cl is the small strain and E, (the initial slope of the theoretical curve) can therefore be identified as Young’s modulus of elasticity. It is therefore obtained that

~1 = 32; cos’ @CR,& + k, cm 2@) + Rz(k, + 2k3 cm 2@)]+ P,

OS 0

(32)

where

where

W(R,,R,)

2: =

1000

- 57O(R; + R;).

(36)

Figure 9 shows the curves of u, vs I, with El = 600 psi. It is seen that, over the range 1 < 1, r; 1.35, equation (34) correlates rather well with the experimental results. An experimental curve of constant strain rate 014%/set is also given in the figure, which is steeper than the curve obtained after relaxatiti. Similarly, in the case of a simple tension test,

HAN-CHIN Wu and REN-FENGYAO

6

Its effect on tensile properties of soft tissues has not

11 N lo,o x9‘I 8

however been investigated and such considerations are beyond the scope of this investigation. It suffices to conclude that considerably more research remains to be done in this connection.

4?IS-6 (n5Es&32l1.0

Fig.

9.

1.3

1.2

1.1 EXTENSION

RATIO X,

Stress e, vs extension ratio I,.

in X2 direc# 0 ai = as = 0, Young’s modulus tion can be shown to be a2

E2 = &sin49cos2

2@

4k;-k:

Acknowledgements-The authors wish to express their sincere appreciation to Professor K. C. Valanis for his valuable comments; to Prof. K. Rim and Dr. C. B. Larson for their support throughout the investigation; to Dr. E. F. Rose and Dr. W. Osebold for their assistance in the preparation of the specimens; and to Dr. M. A. Khowassah for his permission to use the Instron machine. This investigation was supported by Biomedical Sciences Support Grant FR-07035 from the General Research Support Branch, Division of Research Resources, Bureau of Health Professions Education and Manpower Training National Institutes of Health, through the Office of the Vice-President for Educational Development and Research, The University of Iowa.

[k5 cos 29 + k,( 1 + cos2 2Cg)]’ REFERENCES

(37) For the human lumbar ammlus fibrosus, E2 = 530 psi. It is remarked that the initial slopes E, and E, as determined here are greater than those reported by Galante (1967) for ammlus fibrosus. They are also greater than the corresponding theoretical values obtained by Blatz et al. (1969) for animal tissues. This discrepancy is due to the difhculty associated with the definition of a reference state as was discussed at length by Fung (1967). The reference state in this paper is that state where the specimen begins to transmit load. 4.

CONCLUDING REMARKS

It is generally accepted that the deformation of soft tissues involves both fiber slipping motion in the matrix and fiber elongation. Due to the gradual transition from the stage of pure fiber slipping motion to the stage of fiber elongation, as the specimen deforms, it is difficult to determine the reference state exactly, and as a consequence, also difficult to determine the initial slopes El and E2 of the stress-strain curves of soft tissues. This is believed to be. a major factor that causes the discrepancy among the published data on the deformation of soft tissues. In the experiments reported herein the specimens failed when the extension ratio I, was between 1.20 and 1.35 (with an average of 1.25). Thus, the theoretical results were compared to the experimental data over the range 1 I I, I 1.35 (Fig. 9). The specific form of the strain energy function given in equation (36) is therefore valid only within this range of deformation and for the material specified. The average values of the tensile strength was found to be 550 psi. This value and that of the extension ratio Ai are comparable to those found by Galante (1967), although fiber orientations in Galante’s specimens were different than the present ones. The technique of gelatine embedment has been applied in research dealing with biological systems.

Adkins, J. E. and Rivlin. R. S. (1955) Large elastic deformations of isotropic materials. Phil. Trans. R. Sec. (A)24& 201-223. Adkins, J. E. (1956) Cylindrically symmetrical deformations of incompressible elastic materials reinforced with inextensible cords. Rat. Me&. Anal. 5, 189-201. Blatz, P. J., Chu. B. M. and Wayland, H. (1969) On the mechanical behavior of elastic animal tissue. Trans. Sot. Rheol. 13(l), 83-102. Crisp, J. D. C. (1968) Biomechanics-notes of a graduate seminar series. Ind. Inst. SC.. Dept. Aeron. Engng, Rept. No. AE223S. Fung, Y. C. B. (1967) Elasticity of soft tissues in simple elongation Am. J. Physiol. 213, 1532-1544. Galante. J. 0. (19671 Tensile orooerties of the human lumbar annulus ‘fibrosus. Acta’Orihop. Stand. Suppl. 100. Green, A. E. and Adkins, J. E. (1960) Large Elastic Defirmation. Clarendon Press, Oxford. Harkness, R. D. (1966) Rheological problems of collagenous tissues. Lab. Pratt. lS(2). 166170, 183. Hart-Smith, L. J. and Crisp. J. D. C. (1967) Large elastic deformation of thin rubber membranes. Jnt. J. Eng. Sci. 5, l-24. Hildebrandt, J., Fukaya. H. and Martin. C. J. (1968) Completing the length-tension curve of tissue. J. Biomechanics 2, 463-467.

Horton, W. (1958) Further observations on the elastic mechanism of the intervertebral disc. J. Bone. Jnt Surg. (B).lO, 552-557. Kydoniefs. A. D. (1970) Finite axisymmetric deformations of an initially cylindrical membrane reinforced with inextensible cords. Q. JI Mech. appl. Math. 23, 481-488. Lee, J. S.. Frasher. W. G. and Fung. Y. C. B. (1967) Twodimensional finite-deformation on experiments on dog’s arteries and veins. Tech. Rept. No. AFOSR 67-1980. University of Calif., San Diego. Moonev. M. (19401The theorv of larae elastic deformation. J. ap;l. Ph;s. Ii, 582-592: I Rivlin, R. S. (1960) Some topics in finite elasticity. In Structure Mechanics (Edited by Goodier, J. N. and Hoff, N. J.). Pergamon Press, Oxford. Rivlin. R. S. (1964) Networks of inextensible cords. In Nonlinear Problems of Engineering (Edited by Ames, W. F.), pp. 51-64. Academic Press, New York. Rivlin, R. S. (1970) An introduction to non-linear continuum mechanics. In Non-Linear Continuum Theories in Mechanics and Physics and Their Applications. Ediioni Cremonses, Rome.

Mechanical behavior of human annulus fibrosus Saunders, D. W. (1964) Large deformations in amorphous polymers. In Biomechanics and Related Engineering Topits (Edited by Kenedi, R. M.) Pergamon Press. Oxford. Spencer. A. J. M. (1972) Deformations of Fihre-Reinforced Materials. Oxford Science Research Papers.

Valanis. K. C. and Handel. R. I. (1967) The strain-energy function of hyperelastic material in terms of the extension ratios. J. appl. Phys. 38, 2997-3002. Veronda. D. R. and Westmann. R. A. (1970) Mechanical characteristics of skin finite deformation. J. Biorwchanics 3. 111-124.

7