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A constitutive equation for parallel-fibered elastic tissue
A CONSTITUTIVE PARALLEL-FIBERED ROLFF B. Jr~~r~st
EQUATION FOR ELASTIC TISSUE* and ROBERT WM. Lrr-n E
Bettis Atomic Power Laboratory, Westinghouse Electric Company. Department of Mechanical Engineering, Michigan Slate University.
Pittsburgh, Pennsylvania. U.S.A. and East Lansing, Michigan 48823. U.S.A.
constitutive equation for parallel-fibered elastic tissue is developed in the form ofa non-linear hereditary integral. Material constants were determined from relaxation test< and these were used in the constitutive equation to predict results of constant strain rate tests, hysteresis loops and sinusoidal tests. Test specimens were obtained from the ligamentum nuchae of gross bovine specimens.
Abstract-A
INTRODLCTION
Elastic fibers are thin thread-like strands which are composed of a progression of sub-units reaching a characteristic filament observable only by electron microscope. The elastic fiber will be defined here as a strand of cu. l-10 kern in dia. which has as its primary component the protein elastin. The elastic fiber may have collagen fibrils interwoven with the principal component of elastic fibrils. By this definition of fiber size. elastic membranes such as those in the vascular walls and in the lung are also composed of elastic fibers. Elastic tissue may refer to a connective tissue which has elastic fibers as its principal component. Elastic tissue totally devoid of collagenous tissue and other tissues is called elastica. Thus, elastic tissue or elastic fiber has a structural connotation while elastin refers to the chemical character of elastic fibers. Elastic fibers arranged in parallel bundles. flat bands, or membranes always form a continuous network with no free ends. In the state of relaxation. elastic fibers are highly refractile, almost isotropic. and somewhat kinked in appearance. A fine structure description of an elastic fiber has been given by Gross (1949, 1950). By electron microscopy, he was able to define two distinct chemical and morphological components; threads and an amorphus ground substance. The threads are arranged in long, generally parallel bundles which combine in great numbers to form a fiber. The fibers are infiltrated by and imbedded in a trypsin-sensitive ground substance. Collagen filaments are present within the elastic fiber as an incidental component probably incorporated into the fiber extra cellularly during its formulation. The gound substance is apparently responsible for the heat resistance of elastic fibers because when trypsin disperses the matrix of ground substance, the elastic fiber becomes heat labile. Later work by Slack ( 1959) and Porter (1966) do not
the functions of transport, defense, storage. repair and general support. The intercellular material in this tissue is the major component which carries load during the performance of these functions. Therefore, the understanding of the mechanical properties requires the study of the intercellular material. This material may consist of collagenous, elastic or recticular fibers and varying amounts of an amorphous substance. The study of each of these fibers and the base matrix must precede the complete mathematical modeling of connective tissue. Fung (1968) proposed a quasi-linear stress-strainhistory law for biological tissues. Haut and Little (1969) used this form for the constitutive equation for a canine anterior cruciate ligament and later (1972) used a similar form for collagen fibers. This mathematical form uses two dynamic parameters established by relaxation tests and quasi-static tests to characterize the behavior of the material in uniaxial tests. Of the intercellular materials, collagen is the best understood and has received the most study; however, the primary interest here is the elastic fiber. The elastic fiber is usually compared with the collagen fiber because the two are very closely associated anatomically. morphologically. chemically and physiologicallv. Elastic fibers are always found in close anatomic association with collagen fibers and are usually dominated by collagen fibers. Hence elastic fibers are very difficult to isolate and test mechanically. The elastic fiber has a greater extensibility and a much lower tangent modulus in quasi-static tests than a collagen fiber of the same size. Connective
tissue
performs
t Part of this paper 1s taken from the Ph.D. Thesis of the first author. submitted to Michigan State I’ni\cr
398
R. B. JENKINSand R. WM.
confirm all of the observations of Gross. These contradictions imply that the elastic fiber microscopic morphology warrants further investigation. From an anatomic point of view, there appear to be five functions for elastic fiber networks. First, fiber networks disseminate stresses originating at isolated points. Second, networks are designed for coordination of the rhythmic motions of the body parts. Third, fiber networks conserve energy by the maintenance of tone during the relaxation of muscle elements. Fourth, the continuous nature of fibrous networks provides a defense against excessive forces. Fifth, fiber extensibility and retractility assist organs in returning to their undeformed configuration once all forces have been removed. The testing of rheological properties of elastic tissue has most commonly used specimens from the human aorta, the bovine aorta or the bovine ligamentum nuchae. The aortas are ca. 35-40 per cent elastin by dry weight and the bovine ligamentum muchae is 78-83 per cent elastin by the same measure. Roy (1880) tested human aortic strips and provided the first accurate force-extension curves for an elastic fiber-dominant tissue. Krafka (1937) calculated average Young’s moduli for aorta and ligamenturn nuchae of 0.286 x 10“ dyn/cm’ and 3.06 x 10’ dyn/cm’ respectively. Hass (1942, 1943) tested human aortic rings and introduced the variables of age and degree of purification of the tissue. He defined purified fibers as those treated with warm formic acid for 72 hr. Elastica showed 32 per cent greater strain and 170 per cent greater retractility at a given stress than untreated rings. Wood (1954) conducted tests on the ligamentum nuchae at strains up to 20 per cent, avoiding time or hysteresis effects. Wood used data obtained from the native ligament to serve as a control to determine changes upon the load-elongation curve due to various enzymes, acids and testing environments. He concluded that the collagen fibers did not participate to any extent until strains greater than 20 per cent were experienced. Hoeve and Flory (195X) analyzed the thermalmechanical nature of the ligamentum nuchae and concluded that collagen fibers had a significant effect on the load-elongation curve only at strains greater than 70 per cent, Carton, Dianauskas and Clark (1962) proposed a curve fit for single elastic fibers in uniaxial tension. These fibers were teased from fresh ligamentum nuchae and had a negligible quantity of collagen fibers surrounding them. Over a 60 per cent strain range the fibers had an average Young’s modulus of 8.8 x lo6 dyn/cm2. No time effects were observed in strains up to 130 per cent. Larger strips containing some collagen fibers, ground substance and perhaps some reticular fibers gave a lower modulus material with a much dif-
LITTLE
ferent stress-strain curve. They proposed a mathematical model of the form E = l.3(l _ e_O’r’r)
(1)
where .s is the strain and T is the load in dyn. Linear viscoelastic models have been tried for body tissues by Frisen rt al. (1969) and Apter (1967). However, the quasi-linear approach of Fung (1972) seems to offer the most promise. Fung postulates the existence of a relaxation function K(1, t) of the form K(I, t) = G(t) T”(i),
G(0) =
1,
(2)
where 1 is the extension ratio and r<(n) is called the elastic response. Assuming the separable nature of the relaxation function along with the principle of superposition of infinitesimal responses leads to a constitutive equation of the form:
‘;
a(t) = * G(t - 6) s0
[e(S)]
9 (t) d6
(3)
G(t) is called the ‘reduced’ relaxation function and because &(E) is not linear in strain, the above theory is called ‘quasi-linear’.
EXPERIMENTAL
METHODS
In order to determine a uniaxial stress-strain law for the ligamenturn nuchae. three different forms of loading were employed. Low strain rate tests and stress relaxation tests at various levels were conducted at different temperatures. By loading and unloading specimen strips at constant strain rates, hysteresis loops were obtained. Sinusoidal tests at different strain levels and frequencies were performed to verify the form of the constitutive equation. The first test and a quasi-static law were used to determine a uniaxial constitutive equation for elastic tissue and the last two tests to refine this equation. Specimens for testing were obtained from available Holstein cattle at the diagnostic laboratory of the College of Veterinary Medicine. Ligamentum nuchae specimens of 12-15 in. in length and varying cross-sections were available within 8 hr after death. Specimens were immediately washed in cold tap water and cut with scissors into smaller sections which were more suitable for further slicing. These sections were cut with the assistance of two piano wire gratings into strips of ca. 0.75 x 1.5 mm for testing. Strips were placed in a closed container of Lock’s saline solution and stored at 46°C. All tests were conducted with tissue strips immersed in a temperature controlled saline bath. The specimens were held in clamping type grips and were thin enough
Fibered
elastic
that severe squashing was avoided. The gage length of the specimen was 15-18 mm. All specimen dimensions were measured with a traveling microscope. Five measurements of cross-section were taken along the gauge length of the specimen and the cross-section was taken to be the average of these five readings. Two separa,te testing fixtures were used for conducting the tests. The first fixture, used for generating constant strain rates. is similar to many screw-driven tensile testing machines, This machine consists of a crosshead driven by the turning of two finely threaded screws mounted in stainless steel ball bearings. Inserted into the stationary upper plate of the tensile testing fixture is a Statham Model UC3 Universal Transducing Ccl1 along with Statham UL4-0.5 load cell accessory. This cell is powered by a Statham SC1001 Universal Readout Unit and the load cell, accessory, and readout unit can accommodate slightly over 100 g load in tension. Displacements of the crosshead were measured with a steel clip gage inserted between the crosshead and load cell plate. The clip gage was constructed from steel shim stock 0.01 in. in thickness. 0.20 in. in width, and about 6.0 in. in length. Attached to the convex side of the clip gage was a Budd Metalfilm Strain Gauge. Type Cl5624. The strain gauge was balanced in the bridge circuit by a Heath Decade Box Resistance. The clip gauge and variable resistor were used in conjunction with an Ellis Associates Model BAM- 1 Bridge Amplifier Meter. The other testing device was a variable amplitude scotch yoke used for sinusoidal tests. This fixture was used to induce displacements up to 15 mm. Both testing fixtures were powered by a l/70 horsepower Bodine Model NSR-12R shunt wound motor. The speed of the motor was controlled by a Minarik Model W-14 speed control unit. In order to obtain both high and low rates of crossbar displacement without radically changing motor speeds. a gearbox was used to achieve 5:1, 3:l. and I:1 speed reduction ratios. The loads and displacements of all specimens were recorded on a Sanborn 322 Dual Channel DC Amplifier Recorder. Occasionally. constant strain rate loading and unloading tests were monitored on a Varian F-80 X-Y Recorder and at times. both recorders were engaged in parallel. All tests on specimens from a given animal were run during a 4-day or 96-hr period. Literature concerning the effects of time and manner of post-mortal storage on connective tissue mechanical properties (Viidik and Lewin, 1966; Van Brocklin and Ellis, 1965) is contradictory. However, preliminary testing done on the hgamentum nuchae specimens stored in the refrigerator in preparation for this work implied that the mechanical character
of spccimcns
~~1s not
significantI\
altered
399
tissue
within 96 hr after the gross ligament was taken from the cow. When stored from 4 to 7 days, the time dependent characteristics of the specimens were altered although their general ‘elastic’ or stiffness character seemed unaffected. After 7 days storage, specimens began to putrify very noticeably. In order to better preserve the specimens, the saline solution in which they were stored in the refrigerator was changed and the specimens rinsed twice daily.
COLLAGEN
AND
GROUND
SUBSTANCE
EXTRACTION
The mechanical character and the mathematical description of that character of any tissue depends on three factors; the deformational characteristics of the tissue fibers, the proportion of the tissue fibers within the composite tissue and, finally, the geometrical relationship of the tissue fibers with each other and with other components (Crisp, 1972). Since the elastic fibers very strongly dominate all other components of the ligamentum nuchae, it is felt that the mechanical nature of the ligamentum nuchae is very closely related to the mechanical properties of the elastic fibers composing it. However. the effect of collagen fibers, reticular fibers, and ground substance upon the mechanical properties of the iigamentum nuchae cannot be ignored. Some work has been directed toward determining the effects which various components of the ligamentum nuchae have on its mechanical properties (Wood, 1954; Burton, 1954; Partington and Wood. 1963). There is generally little disagreement among these works as to what these effects are in a qualitative sense. An effort was made to determine how the components of the ligamenturn nuchae effect the quantitative mathematical model of the composite tissue. The most certain way of determining the properties of the components ofany tissue is to isolate the various components by some physical or chemical means. In the case of the ligamentum nuchae. physical isolation of components for mechanical testing has proved infeasible. Chemical methods have proved more successful but the results of chemical treatments must be studied histologically to determine their effectiveness. Elastic fibers have been removed from the ligament by various methods which have different effects upon the ground substance. The effectiveness of these methods was studied by prepared slides which also provided information concerning the structural arrangement of the lipamentum nuchae in different stages of purification. Boiling water removes young collagen fibers and ground substance from the ligamentum nuchae very effectively. Prolonged exposure to concentrated (XXper
R. B. JENKINSand R. WM. LITTLE
400
cent) formic acid at 4550°C is the most effective means of removing collagen and ground substance. Both methods were employed in this study. The ligamenturn nuchae of an animal was cut into four separate sections for four different preparations. One section was heated in distilled water for 90 min at 58°C and then stored in a refrigerator. The second section was boiled in distilled water for 7.5 hr and refrigerated. The third section was heated in 88 per cent formic acid for 24 hr in a temperature controlled oven at 48°C and refrigerated. The fourth section was not treated in any manner. Refrigeration times of all sections were the same and stress relaxation tests were conducted on samples from each section. Pieces from the untreated, boiled and formic acid treated sections were used to prepare histological slides. Three stains, Hematoxylin-Eosin, Gomori’s one step trichrome, and Verhoeff, were used for contrast of displaying either collagen or elastic fibers. The effects of chemical purification on structure were examined and compared to changes in mechanical characteristics. The histological slides indicated that all collagen was removed by the formic acid treatment and only a small amount of reticular material remained after boiling. These slide studies correlated well with the mechanical test data.
Haut and Little (1972) noted that the elastic response of collagen was approximately described by the second term of the series shown in equation (5). The first two terms were assumed as a first approximation to elastic tissue. Previous stress relaxation tests on biological material have shown that stress relaxes with the logarithm of time (Apter, 1967; Rigby et al., 1939; Abrahams, 1967). In preliminary tests, this was shown to be true for ligamentum nuchae for a given level of strain. However, the slope of the curve was experimentally determined to be dependent upon the square of the strain and the relaxation function, G, was assumed to be: G(E,t) = 1 + p’(t) In t
(6)
where t is a normalized time (r/7’). Equation (6) implies that relaxation is linearly dependent upon the logarithm of time only at a specified strain. The generalized non-linear hereditary integral was taken as a(t) =
’ [l + p2(t - i)ln(t - i)] i0
WI + IIC + 2WOl -g-. In stress relaxation the strain and strain rate become
DISCUSSION OF RESULTS
E = co H(t) and zf = co s(t)
Previous experiments by Fung (1967) and later by Haut and Little (1969) indicated that the elastic stress, &‘, in equation (3) could be taken in the form of an exponential function of strain of the form
where H(t) is the unit step function and s(t) is the Dirac delta function. Substituting equation (8) into equation (7) gives the response for stress relaxation tests.
foe= A[1 - l/i.2]e””
0(t) = [Cc, + ofi] [l + & In t].
(4)
or changing from the extension ratio to strain and expanding in a power series: rr’ = Cc + DE'+ ".
(5)
(9)
Low strain rate tests (3 per cent/min) and relaxation tests were run on specimens from four different animals to establish the three ‘material constants’, C, D
,Table 1. Experimental constants
Age (yr) 05 0.5 2 5 Untreated Heated Boiled Acid
* Time is non-dimensionalized
D
C (dyn/cm?
(dyn/cm2)
Ave. a/e (dyn/cm*)
P *
30 40 347 41.9 35 40
4.0 x lo6 4.6 4.4 4.7 3.2 3.0
3.2 x lo6 3.0 1.4 1.1 1.65 2.2
4.7 X 106 5-3 4.9 5.13 3.81 3.83
- 0.0501 -0.0455 - 00247 -0.0188 -00102 - 0.0092
35 35 35 35
2.3 3.25 3.9 2.8
1.85 2.0 0 0
2.85 3.81 3.63 2.12
- 0.0247 - 0.0208 0 0
Temperature (“C)
by characteristic time of 1 min.
401
Fibered elastic tissue (9) was found to agree with experimental observations in that relaxation is linear with logarithm of time after small times (t5 set). The four animals used as sources for test samples varied in age from 6 months to 5 yr. Two were young animals (6 months), one was 2 yr old and one was 5 yr old. The effects of chemical purification were examined only on the 5 yr animal. The constants were determined for each of these specimens and are in Table 1. The temperature of the test bath was varied and results are the average of 8- 10 tests on each specimen. Quasistatic tests were run to establish the constants C and D and the slope of the relaxation curve yielded p. These results indicate that the C constant carries most of the dependency of the elastic fibers and the non-linear and time effects, D and p. are most dependent upon the collagen present. In all relaxation tests. no relaxation was observed for constant strain levels below 30 per cent. This indicated that the observed wavyness in collagen in connective tissue is such that the collagen does not take load until higher strains are reached. The ‘quasi-elastic’ response of the last four specimen groups may be seen in Fig. 1. For constant strain rate tests, the strain may be written and p. Equation
E =
Pt.
(‘I)
Substituting this into equation (7) yields
% Fig. 2.
Strain
Strain rate effects
Hysteresis effects may be studied by examining loading and unloading at constant strain rates. e(t) =
o
Bt i 2/h, - fit
(13)
t, < t < 2t,
The loading cycle, 0 < t < t,, is given by equation (12) and the unloading becomes,
(12) - 2/&t
@-
Theoretical
-- -
a(t) = CE + De’
a(r) = CE + De2 + @1 [[C’+~~ln;~:[C’+~D~].
73% /m1n -
- t,)[[c[2t:
Heated
- t,(t - t,)
+ f(t - t, ,“I
@>- Boiled @-
Untreated
a,-
Formic
mid -treated
+ 4DPt, [Zrf - t,t + i(t - tl)‘Jl
_
c
ii
2t*
_
I
t,
(t - tl) I 0 - t,)’ 2
+ 4D/3t, 2t”, - t,t + ;(t
%
Strain
Fig. 1. Stress-strain plot.
ln(t-f,)
9 - t,)’
I
III
(14)
The ligamentum nuchae does not display strong hysteresis effects. A comparison between the theory (equations 12and 14) and experiments is shown in Fig. 2. The final test was a sinusoidal strain test at different frequencies. The strain input was of the form E = E,/2( 1 - cos 0.It)
(15)
402
R. B. JENKINS and R. WM.
where la is the strain amplitude and w the frequency. The simplest method of examining the experimental data was to determine the stress amplitude decay for times when the strain was a maximum. Equation (12) predicts a stress amplitude of the form:
LITTLE
Fung, Y. C. B. (1968) Biomechanics (its scope, history and some problems of continuum mechanics in physiology). Appl. Me&. Reu. 21 (I), l-20. Fung, Y. C. B. (1972) Stress-strain history relations of soft tissues in simple elongation. In Biornechanics: Its Foundations arid Objectives (Edited by Fung, Y. C. B., Perrowe, -
&I? ~J(~)=CE,+&,~+~(C+DE~)
-~y+~lnt+~Ci(ot) [
+/&D L$ g
-~
1
ln t - Ci(wt) + i Ci(2wt) Ci(4ot)
Ci(3wt) 3
Ci(2wt) -__ + Ci(bt) 16 2
+,
where y (Euler’s constant) = 0577 and Ci(x) is the cosine integral function. Equation (16) predicts a stress amplitude decay at a reduced rate from that observed in relaxation tests. This theoretical result was not observed in any of the experimental tests and the stress amplitude relaxed at approximately the same rate as the stress relaxation in constant strain tests. CONCLUSIONS
A quasi-linear viscoelastic law incorporating a relaxation function and a non-linear elastic response has been formulated. Treatment of the tissue to remove the collagen resulted in a non-linear elastic material. The results of this work together with results of investigations of collagen fibers may now serve as the basis for composite material approaches to soft connective tissues. REFERENCES
16
(16) 1 N. and Anliker. M.), pp. 181-208. wood Cliffs. N.J.
Prentice
Hall,
Engle-
Gross, J. (1949) Structure of elastic tissue as studied by the electron microscope. J. Exp. Med. 89, 699-707. Gross, J. (1950) Connective tissue fine structure and some methods of its analysis. J. Geronrol. 5, 343-356. Hass, G. M. (1942) Elastic tissue-II. A study of the elasticity and tensile strength of elastic tissue isolated from the human aorta. A&. Parkol. 34, 971-981. Hass, G. M. (1943) Elastic tissue-III. Relations between the structure of the aging aorta and the properties of the isolated aortic elastic tissue. Arch. Puthol. 35, 29-45. Haut, R. C. and Little, R. W. (1969) The rheological properties of canine anterior cruciate ligaments. J. Biomechanics 2,289-298.
Haut, R. C. and Little, R. M. (1972) A constitutive equation for collagen fibers. J. Bionlechanics 5,423-430. Hoeve, C. A. J. and Flory, P. J. (1958) The elastic properties of elastin. d. Am. Chem. Sot. 80, 6523-6526. Krafka, J., Jr. (1937) Mechanical factors in arteriosclerosis. Arch. Pathol. 23, l-14. Partington, F. R. and Wood, G. C. (1963) The role of noncollagen components in the mechanical behavior of tendon fibers. Biochim. Biophys. Acta 69,485-495.
Porter, K. R. (1966) Mesenchema and connective tissue. In Abrahams, M. (1967) Mechanical behavior of tendon in vitro. Med. Biol. Engng 5, 433-443.
Apter, J. T. (1967) Correlation of viscoelastic properties with. microscop.ic structure oflarge arteries. Circ. Res. 21, 901918. Burton. A. C. (1954) Relation of structure to function of the tissues of the wall of blood vessels. Phw. Reu. 34, 619-642. Carton, R. W., Dainauskas, J. and Clark; J. W. (1962) Elastic properties of single elastic fibers. J. appl. Physiol. 17, 547551. Crisp, J. D. C. (1972) Properties of tendon and skin. In Biomechanics: Its Foundations and Ohjrctiues (Edited by Fung, Y. C. B., Perrowe, N. and Anliker, M.), pp. 141179. Prentice Hall, Englewood Cliffs, N.J. Frisen, M., Magi, M., Sonnerup, L. and Viidik. A. (1969) Rheological analysis of soft collagenous tissue. J. Biomechanics f, 13-Z. _ Fung, Y. C. B. (19673 Elasticity of soft tissues in simple elongation. .4/n. J. Ph~aiol. 213 (6), 1532-l 544.
Histology (Edited by Greep, R. O.), 2nd Edn., pp. 99-133. McGraw Hill, New York. Rigby, B., Hirai, N., Spikes, J. and Eyring, H. (1959) The mechanical properties of rat tail tendon. J. Gen. Physiol. 43,265-283. Roy, C. S. (1880) Elastic properties of the arterial wall. J. Physiol. 3, 125-159. Slack, H. G. B. (1959) Some notes on the composition of metabolism of connective tissue. Amer. J. Med. 26, 113124. Van Brocklin, J. D. and Ellis, D. (1965) A study of the mechanical behavior of toe extensor tendons under applied stress. Arch. Phys. Med. Rehab. 46, 369-373. Viidik, A. and Lewin, T. (1966) Changes in the tensile strength characteristics and histology of rabbit ligaments induced by different modes of post-mortal storage. Acta Orthop. Stand. 37, 141-155. Wood, G. C. (1954) Some tensile properties of elastic tissues. Biochirn. Biophys. Acta 15, 31 l-324.
EQUATION FOR ELASTIC TISSUE* and ROBERT WM. Lrr-n E
Bettis Atomic Power Laboratory, Westinghouse Electric Company. Department of Mechanical Engineering, Michigan Slate University.
Pittsburgh, Pennsylvania. U.S.A. and East Lansing, Michigan 48823. U.S.A.
constitutive equation for parallel-fibered elastic tissue is developed in the form ofa non-linear hereditary integral. Material constants were determined from relaxation test< and these were used in the constitutive equation to predict results of constant strain rate tests, hysteresis loops and sinusoidal tests. Test specimens were obtained from the ligamentum nuchae of gross bovine specimens.
Abstract-A
INTRODLCTION
Elastic fibers are thin thread-like strands which are composed of a progression of sub-units reaching a characteristic filament observable only by electron microscope. The elastic fiber will be defined here as a strand of cu. l-10 kern in dia. which has as its primary component the protein elastin. The elastic fiber may have collagen fibrils interwoven with the principal component of elastic fibrils. By this definition of fiber size. elastic membranes such as those in the vascular walls and in the lung are also composed of elastic fibers. Elastic tissue may refer to a connective tissue which has elastic fibers as its principal component. Elastic tissue totally devoid of collagenous tissue and other tissues is called elastica. Thus, elastic tissue or elastic fiber has a structural connotation while elastin refers to the chemical character of elastic fibers. Elastic fibers arranged in parallel bundles. flat bands, or membranes always form a continuous network with no free ends. In the state of relaxation. elastic fibers are highly refractile, almost isotropic. and somewhat kinked in appearance. A fine structure description of an elastic fiber has been given by Gross (1949, 1950). By electron microscopy, he was able to define two distinct chemical and morphological components; threads and an amorphus ground substance. The threads are arranged in long, generally parallel bundles which combine in great numbers to form a fiber. The fibers are infiltrated by and imbedded in a trypsin-sensitive ground substance. Collagen filaments are present within the elastic fiber as an incidental component probably incorporated into the fiber extra cellularly during its formulation. The gound substance is apparently responsible for the heat resistance of elastic fibers because when trypsin disperses the matrix of ground substance, the elastic fiber becomes heat labile. Later work by Slack ( 1959) and Porter (1966) do not
the functions of transport, defense, storage. repair and general support. The intercellular material in this tissue is the major component which carries load during the performance of these functions. Therefore, the understanding of the mechanical properties requires the study of the intercellular material. This material may consist of collagenous, elastic or recticular fibers and varying amounts of an amorphous substance. The study of each of these fibers and the base matrix must precede the complete mathematical modeling of connective tissue. Fung (1968) proposed a quasi-linear stress-strainhistory law for biological tissues. Haut and Little (1969) used this form for the constitutive equation for a canine anterior cruciate ligament and later (1972) used a similar form for collagen fibers. This mathematical form uses two dynamic parameters established by relaxation tests and quasi-static tests to characterize the behavior of the material in uniaxial tests. Of the intercellular materials, collagen is the best understood and has received the most study; however, the primary interest here is the elastic fiber. The elastic fiber is usually compared with the collagen fiber because the two are very closely associated anatomically. morphologically. chemically and physiologicallv. Elastic fibers are always found in close anatomic association with collagen fibers and are usually dominated by collagen fibers. Hence elastic fibers are very difficult to isolate and test mechanically. The elastic fiber has a greater extensibility and a much lower tangent modulus in quasi-static tests than a collagen fiber of the same size. Connective
tissue
performs
t Part of this paper 1s taken from the Ph.D. Thesis of the first author. submitted to Michigan State I’ni\cr
398
R. B. JENKINSand R. WM.
confirm all of the observations of Gross. These contradictions imply that the elastic fiber microscopic morphology warrants further investigation. From an anatomic point of view, there appear to be five functions for elastic fiber networks. First, fiber networks disseminate stresses originating at isolated points. Second, networks are designed for coordination of the rhythmic motions of the body parts. Third, fiber networks conserve energy by the maintenance of tone during the relaxation of muscle elements. Fourth, the continuous nature of fibrous networks provides a defense against excessive forces. Fifth, fiber extensibility and retractility assist organs in returning to their undeformed configuration once all forces have been removed. The testing of rheological properties of elastic tissue has most commonly used specimens from the human aorta, the bovine aorta or the bovine ligamentum nuchae. The aortas are ca. 35-40 per cent elastin by dry weight and the bovine ligamentum muchae is 78-83 per cent elastin by the same measure. Roy (1880) tested human aortic strips and provided the first accurate force-extension curves for an elastic fiber-dominant tissue. Krafka (1937) calculated average Young’s moduli for aorta and ligamenturn nuchae of 0.286 x 10“ dyn/cm’ and 3.06 x 10’ dyn/cm’ respectively. Hass (1942, 1943) tested human aortic rings and introduced the variables of age and degree of purification of the tissue. He defined purified fibers as those treated with warm formic acid for 72 hr. Elastica showed 32 per cent greater strain and 170 per cent greater retractility at a given stress than untreated rings. Wood (1954) conducted tests on the ligamentum nuchae at strains up to 20 per cent, avoiding time or hysteresis effects. Wood used data obtained from the native ligament to serve as a control to determine changes upon the load-elongation curve due to various enzymes, acids and testing environments. He concluded that the collagen fibers did not participate to any extent until strains greater than 20 per cent were experienced. Hoeve and Flory (195X) analyzed the thermalmechanical nature of the ligamentum nuchae and concluded that collagen fibers had a significant effect on the load-elongation curve only at strains greater than 70 per cent, Carton, Dianauskas and Clark (1962) proposed a curve fit for single elastic fibers in uniaxial tension. These fibers were teased from fresh ligamentum nuchae and had a negligible quantity of collagen fibers surrounding them. Over a 60 per cent strain range the fibers had an average Young’s modulus of 8.8 x lo6 dyn/cm2. No time effects were observed in strains up to 130 per cent. Larger strips containing some collagen fibers, ground substance and perhaps some reticular fibers gave a lower modulus material with a much dif-
LITTLE
ferent stress-strain curve. They proposed a mathematical model of the form E = l.3(l _ e_O’r’r)
(1)
where .s is the strain and T is the load in dyn. Linear viscoelastic models have been tried for body tissues by Frisen rt al. (1969) and Apter (1967). However, the quasi-linear approach of Fung (1972) seems to offer the most promise. Fung postulates the existence of a relaxation function K(1, t) of the form K(I, t) = G(t) T”(i),
G(0) =
1,
(2)
where 1 is the extension ratio and r<(n) is called the elastic response. Assuming the separable nature of the relaxation function along with the principle of superposition of infinitesimal responses leads to a constitutive equation of the form:
‘;
a(t) = * G(t - 6) s0
[e(S)]
9 (t) d6
(3)
G(t) is called the ‘reduced’ relaxation function and because &(E) is not linear in strain, the above theory is called ‘quasi-linear’.
EXPERIMENTAL
METHODS
In order to determine a uniaxial stress-strain law for the ligamenturn nuchae. three different forms of loading were employed. Low strain rate tests and stress relaxation tests at various levels were conducted at different temperatures. By loading and unloading specimen strips at constant strain rates, hysteresis loops were obtained. Sinusoidal tests at different strain levels and frequencies were performed to verify the form of the constitutive equation. The first test and a quasi-static law were used to determine a uniaxial constitutive equation for elastic tissue and the last two tests to refine this equation. Specimens for testing were obtained from available Holstein cattle at the diagnostic laboratory of the College of Veterinary Medicine. Ligamentum nuchae specimens of 12-15 in. in length and varying cross-sections were available within 8 hr after death. Specimens were immediately washed in cold tap water and cut with scissors into smaller sections which were more suitable for further slicing. These sections were cut with the assistance of two piano wire gratings into strips of ca. 0.75 x 1.5 mm for testing. Strips were placed in a closed container of Lock’s saline solution and stored at 46°C. All tests were conducted with tissue strips immersed in a temperature controlled saline bath. The specimens were held in clamping type grips and were thin enough
Fibered
elastic
that severe squashing was avoided. The gage length of the specimen was 15-18 mm. All specimen dimensions were measured with a traveling microscope. Five measurements of cross-section were taken along the gauge length of the specimen and the cross-section was taken to be the average of these five readings. Two separa,te testing fixtures were used for conducting the tests. The first fixture, used for generating constant strain rates. is similar to many screw-driven tensile testing machines, This machine consists of a crosshead driven by the turning of two finely threaded screws mounted in stainless steel ball bearings. Inserted into the stationary upper plate of the tensile testing fixture is a Statham Model UC3 Universal Transducing Ccl1 along with Statham UL4-0.5 load cell accessory. This cell is powered by a Statham SC1001 Universal Readout Unit and the load cell, accessory, and readout unit can accommodate slightly over 100 g load in tension. Displacements of the crosshead were measured with a steel clip gage inserted between the crosshead and load cell plate. The clip gage was constructed from steel shim stock 0.01 in. in thickness. 0.20 in. in width, and about 6.0 in. in length. Attached to the convex side of the clip gage was a Budd Metalfilm Strain Gauge. Type Cl5624. The strain gauge was balanced in the bridge circuit by a Heath Decade Box Resistance. The clip gauge and variable resistor were used in conjunction with an Ellis Associates Model BAM- 1 Bridge Amplifier Meter. The other testing device was a variable amplitude scotch yoke used for sinusoidal tests. This fixture was used to induce displacements up to 15 mm. Both testing fixtures were powered by a l/70 horsepower Bodine Model NSR-12R shunt wound motor. The speed of the motor was controlled by a Minarik Model W-14 speed control unit. In order to obtain both high and low rates of crossbar displacement without radically changing motor speeds. a gearbox was used to achieve 5:1, 3:l. and I:1 speed reduction ratios. The loads and displacements of all specimens were recorded on a Sanborn 322 Dual Channel DC Amplifier Recorder. Occasionally. constant strain rate loading and unloading tests were monitored on a Varian F-80 X-Y Recorder and at times. both recorders were engaged in parallel. All tests on specimens from a given animal were run during a 4-day or 96-hr period. Literature concerning the effects of time and manner of post-mortal storage on connective tissue mechanical properties (Viidik and Lewin, 1966; Van Brocklin and Ellis, 1965) is contradictory. However, preliminary testing done on the hgamentum nuchae specimens stored in the refrigerator in preparation for this work implied that the mechanical character
of spccimcns
~~1s not
significantI\
altered
399
tissue
within 96 hr after the gross ligament was taken from the cow. When stored from 4 to 7 days, the time dependent characteristics of the specimens were altered although their general ‘elastic’ or stiffness character seemed unaffected. After 7 days storage, specimens began to putrify very noticeably. In order to better preserve the specimens, the saline solution in which they were stored in the refrigerator was changed and the specimens rinsed twice daily.
COLLAGEN
AND
GROUND
SUBSTANCE
EXTRACTION
The mechanical character and the mathematical description of that character of any tissue depends on three factors; the deformational characteristics of the tissue fibers, the proportion of the tissue fibers within the composite tissue and, finally, the geometrical relationship of the tissue fibers with each other and with other components (Crisp, 1972). Since the elastic fibers very strongly dominate all other components of the ligamentum nuchae, it is felt that the mechanical nature of the ligamentum nuchae is very closely related to the mechanical properties of the elastic fibers composing it. However. the effect of collagen fibers, reticular fibers, and ground substance upon the mechanical properties of the iigamentum nuchae cannot be ignored. Some work has been directed toward determining the effects which various components of the ligamentum nuchae have on its mechanical properties (Wood, 1954; Burton, 1954; Partington and Wood. 1963). There is generally little disagreement among these works as to what these effects are in a qualitative sense. An effort was made to determine how the components of the ligamenturn nuchae effect the quantitative mathematical model of the composite tissue. The most certain way of determining the properties of the components ofany tissue is to isolate the various components by some physical or chemical means. In the case of the ligamentum nuchae. physical isolation of components for mechanical testing has proved infeasible. Chemical methods have proved more successful but the results of chemical treatments must be studied histologically to determine their effectiveness. Elastic fibers have been removed from the ligament by various methods which have different effects upon the ground substance. The effectiveness of these methods was studied by prepared slides which also provided information concerning the structural arrangement of the lipamentum nuchae in different stages of purification. Boiling water removes young collagen fibers and ground substance from the ligamentum nuchae very effectively. Prolonged exposure to concentrated (XXper
R. B. JENKINSand R. WM. LITTLE
400
cent) formic acid at 4550°C is the most effective means of removing collagen and ground substance. Both methods were employed in this study. The ligamenturn nuchae of an animal was cut into four separate sections for four different preparations. One section was heated in distilled water for 90 min at 58°C and then stored in a refrigerator. The second section was boiled in distilled water for 7.5 hr and refrigerated. The third section was heated in 88 per cent formic acid for 24 hr in a temperature controlled oven at 48°C and refrigerated. The fourth section was not treated in any manner. Refrigeration times of all sections were the same and stress relaxation tests were conducted on samples from each section. Pieces from the untreated, boiled and formic acid treated sections were used to prepare histological slides. Three stains, Hematoxylin-Eosin, Gomori’s one step trichrome, and Verhoeff, were used for contrast of displaying either collagen or elastic fibers. The effects of chemical purification on structure were examined and compared to changes in mechanical characteristics. The histological slides indicated that all collagen was removed by the formic acid treatment and only a small amount of reticular material remained after boiling. These slide studies correlated well with the mechanical test data.
Haut and Little (1972) noted that the elastic response of collagen was approximately described by the second term of the series shown in equation (5). The first two terms were assumed as a first approximation to elastic tissue. Previous stress relaxation tests on biological material have shown that stress relaxes with the logarithm of time (Apter, 1967; Rigby et al., 1939; Abrahams, 1967). In preliminary tests, this was shown to be true for ligamentum nuchae for a given level of strain. However, the slope of the curve was experimentally determined to be dependent upon the square of the strain and the relaxation function, G, was assumed to be: G(E,t) = 1 + p’(t) In t
(6)
where t is a normalized time (r/7’). Equation (6) implies that relaxation is linearly dependent upon the logarithm of time only at a specified strain. The generalized non-linear hereditary integral was taken as a(t) =
’ [l + p2(t - i)ln(t - i)] i0
WI + IIC + 2WOl -g-. In stress relaxation the strain and strain rate become
DISCUSSION OF RESULTS
E = co H(t) and zf = co s(t)
Previous experiments by Fung (1967) and later by Haut and Little (1969) indicated that the elastic stress, &‘, in equation (3) could be taken in the form of an exponential function of strain of the form
where H(t) is the unit step function and s(t) is the Dirac delta function. Substituting equation (8) into equation (7) gives the response for stress relaxation tests.
foe= A[1 - l/i.2]e””
0(t) = [Cc, + ofi] [l + & In t].
(4)
or changing from the extension ratio to strain and expanding in a power series: rr’ = Cc + DE'+ ".
(5)
(9)
Low strain rate tests (3 per cent/min) and relaxation tests were run on specimens from four different animals to establish the three ‘material constants’, C, D
,Table 1. Experimental constants
Age (yr) 05 0.5 2 5 Untreated Heated Boiled Acid
* Time is non-dimensionalized
D
C (dyn/cm?
(dyn/cm2)
Ave. a/e (dyn/cm*)
P *
30 40 347 41.9 35 40
4.0 x lo6 4.6 4.4 4.7 3.2 3.0
3.2 x lo6 3.0 1.4 1.1 1.65 2.2
4.7 X 106 5-3 4.9 5.13 3.81 3.83
- 0.0501 -0.0455 - 00247 -0.0188 -00102 - 0.0092
35 35 35 35
2.3 3.25 3.9 2.8
1.85 2.0 0 0
2.85 3.81 3.63 2.12
- 0.0247 - 0.0208 0 0
Temperature (“C)
by characteristic time of 1 min.
401
Fibered elastic tissue (9) was found to agree with experimental observations in that relaxation is linear with logarithm of time after small times (t5 set). The four animals used as sources for test samples varied in age from 6 months to 5 yr. Two were young animals (6 months), one was 2 yr old and one was 5 yr old. The effects of chemical purification were examined only on the 5 yr animal. The constants were determined for each of these specimens and are in Table 1. The temperature of the test bath was varied and results are the average of 8- 10 tests on each specimen. Quasistatic tests were run to establish the constants C and D and the slope of the relaxation curve yielded p. These results indicate that the C constant carries most of the dependency of the elastic fibers and the non-linear and time effects, D and p. are most dependent upon the collagen present. In all relaxation tests. no relaxation was observed for constant strain levels below 30 per cent. This indicated that the observed wavyness in collagen in connective tissue is such that the collagen does not take load until higher strains are reached. The ‘quasi-elastic’ response of the last four specimen groups may be seen in Fig. 1. For constant strain rate tests, the strain may be written and p. Equation
E =
Pt.
(‘I)
Substituting this into equation (7) yields
% Fig. 2.
Strain
Strain rate effects
Hysteresis effects may be studied by examining loading and unloading at constant strain rates. e(t) =
o
Bt i 2/h, - fit
(13)
t, < t < 2t,
The loading cycle, 0 < t < t,, is given by equation (12) and the unloading becomes,
(12) - 2/&t
@-
Theoretical
-- -
a(t) = CE + De’
a(r) = CE + De2 + @1 [[C’+~~ln;~:[C’+~D~].
73% /m1n -
- t,)[[c[2t:
Heated
- t,(t - t,)
+ f(t - t, ,“I
@>- Boiled @-
Untreated
a,-
Formic
mid -treated
+ 4DPt, [Zrf - t,t + i(t - tl)‘Jl
_
c
ii
2t*
_
I
t,
(t - tl) I 0 - t,)’ 2
+ 4D/3t, 2t”, - t,t + ;(t
%
Strain
Fig. 1. Stress-strain plot.
ln(t-f,)
9 - t,)’
I
III
(14)
The ligamentum nuchae does not display strong hysteresis effects. A comparison between the theory (equations 12and 14) and experiments is shown in Fig. 2. The final test was a sinusoidal strain test at different frequencies. The strain input was of the form E = E,/2( 1 - cos 0.It)
(15)
402
R. B. JENKINS and R. WM.
where la is the strain amplitude and w the frequency. The simplest method of examining the experimental data was to determine the stress amplitude decay for times when the strain was a maximum. Equation (12) predicts a stress amplitude of the form:
LITTLE
Fung, Y. C. B. (1968) Biomechanics (its scope, history and some problems of continuum mechanics in physiology). Appl. Me&. Reu. 21 (I), l-20. Fung, Y. C. B. (1972) Stress-strain history relations of soft tissues in simple elongation. In Biornechanics: Its Foundations arid Objectives (Edited by Fung, Y. C. B., Perrowe, -
&I? ~J(~)=CE,+&,~+~(C+DE~)
-~y+~lnt+~Ci(ot) [
+/&D L$ g
-~
1
ln t - Ci(wt) + i Ci(2wt) Ci(4ot)
Ci(3wt) 3
Ci(2wt) -__ + Ci(bt) 16 2
+,
where y (Euler’s constant) = 0577 and Ci(x) is the cosine integral function. Equation (16) predicts a stress amplitude decay at a reduced rate from that observed in relaxation tests. This theoretical result was not observed in any of the experimental tests and the stress amplitude relaxed at approximately the same rate as the stress relaxation in constant strain tests. CONCLUSIONS
A quasi-linear viscoelastic law incorporating a relaxation function and a non-linear elastic response has been formulated. Treatment of the tissue to remove the collagen resulted in a non-linear elastic material. The results of this work together with results of investigations of collagen fibers may now serve as the basis for composite material approaches to soft connective tissues. REFERENCES
16
(16) 1 N. and Anliker. M.), pp. 181-208. wood Cliffs. N.J.
Prentice
Hall,
Engle-
Gross, J. (1949) Structure of elastic tissue as studied by the electron microscope. J. Exp. Med. 89, 699-707. Gross, J. (1950) Connective tissue fine structure and some methods of its analysis. J. Geronrol. 5, 343-356. Hass, G. M. (1942) Elastic tissue-II. A study of the elasticity and tensile strength of elastic tissue isolated from the human aorta. A&. Parkol. 34, 971-981. Hass, G. M. (1943) Elastic tissue-III. Relations between the structure of the aging aorta and the properties of the isolated aortic elastic tissue. Arch. Puthol. 35, 29-45. Haut, R. C. and Little, R. W. (1969) The rheological properties of canine anterior cruciate ligaments. J. Biomechanics 2,289-298.
Haut, R. C. and Little, R. M. (1972) A constitutive equation for collagen fibers. J. Bionlechanics 5,423-430. Hoeve, C. A. J. and Flory, P. J. (1958) The elastic properties of elastin. d. Am. Chem. Sot. 80, 6523-6526. Krafka, J., Jr. (1937) Mechanical factors in arteriosclerosis. Arch. Pathol. 23, l-14. Partington, F. R. and Wood, G. C. (1963) The role of noncollagen components in the mechanical behavior of tendon fibers. Biochim. Biophys. Acta 69,485-495.
Porter, K. R. (1966) Mesenchema and connective tissue. In Abrahams, M. (1967) Mechanical behavior of tendon in vitro. Med. Biol. Engng 5, 433-443.
Apter, J. T. (1967) Correlation of viscoelastic properties with. microscop.ic structure oflarge arteries. Circ. Res. 21, 901918. Burton. A. C. (1954) Relation of structure to function of the tissues of the wall of blood vessels. Phw. Reu. 34, 619-642. Carton, R. W., Dainauskas, J. and Clark; J. W. (1962) Elastic properties of single elastic fibers. J. appl. Physiol. 17, 547551. Crisp, J. D. C. (1972) Properties of tendon and skin. In Biomechanics: Its Foundations and Ohjrctiues (Edited by Fung, Y. C. B., Perrowe, N. and Anliker, M.), pp. 141179. Prentice Hall, Englewood Cliffs, N.J. Frisen, M., Magi, M., Sonnerup, L. and Viidik. A. (1969) Rheological analysis of soft collagenous tissue. J. Biomechanics f, 13-Z. _ Fung, Y. C. B. (19673 Elasticity of soft tissues in simple elongation. .4/n. J. Ph~aiol. 213 (6), 1532-l 544.
Histology (Edited by Greep, R. O.), 2nd Edn., pp. 99-133. McGraw Hill, New York. Rigby, B., Hirai, N., Spikes, J. and Eyring, H. (1959) The mechanical properties of rat tail tendon. J. Gen. Physiol. 43,265-283. Roy, C. S. (1880) Elastic properties of the arterial wall. J. Physiol. 3, 125-159. Slack, H. G. B. (1959) Some notes on the composition of metabolism of connective tissue. Amer. J. Med. 26, 113124. Van Brocklin, J. D. and Ellis, D. (1965) A study of the mechanical behavior of toe extensor tendons under applied stress. Arch. Phys. Med. Rehab. 46, 369-373. Viidik, A. and Lewin, T. (1966) Changes in the tensile strength characteristics and histology of rabbit ligaments induced by different modes of post-mortal storage. Acta Orthop. Stand. 37, 141-155. Wood, G. C. (1954) Some tensile properties of elastic tissues. Biochirn. Biophys. Acta 15, 31 l-324.