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Passive mechanical properties of uterine muscle (myometrium) tested in vitro
PASSIVE MECHANICAL PROPERTIES OF UTERINE MUSCLE (MYOMETRIUM) TESTED IN VITRO* G. W.
and V. L. ROBERTS
PFARSALL
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27706, U.S.A. Abstract-The mechanical properties of human myometrium in uirro were determined in the passive state at strain rates ofapprox. 0.5 min. Both tension and compression data are presented as curves of true stress vs true strain. The test results represent a range of uterine conditions, from nulliparous to multiparous, in various stages of the menstrual cycle, and include one pregnant uterus. For stresses above 1psi, either tension or compression data can be represented approximately by an equation of the form E = C In u + B. Elastic moduli, measured at an applied stress of 10 psi, were within the range 70-200 psi. and tensile strengths were in _ the range 8&300 psi.
INTRODUCTION
metrium,
The mechanical properties of uterine muscle (myometrium) are of fundamental importance for studying uterine activity during pregnancy and labor, as well as for assessing the susceptibility of the uterus to damage by penetration (Wilson et al., 1977). Yet relatively little stress-strain information has been published describing the mechanical behavior of this muscle, and a number of questions and contradictions exist among the published results. An extensive bibliography has been compiled by Bell (1972), summarizing the mechanical-property studies of myometrium which have been published. Tensile mechanical properties of strips excised from intact uteri have been measured by Csapo and Goodall (1954). Schofield and Wood (1964), Wood (1964a, b), Conrad et al. (1966a, 1966b, 1967), and Mosler (1968). To our knowledge, no compressive stressstrain measurements have been reported for myometrium. Also we are unaware of tensile stress-strain data having been measured from myometrium under conditions of approximately constant strain rate. In the first studies of length-tension relationships in uterine muscle, Csapo and Goodall (1954) excised strips of rabbit uteri and suspended them in saturated Krebs solution by platinum hooks attached to a tensometer, measuring tensile forces produced by electrical stimulation. They plotted normalized force (force/maximum force) vs normalized Length (length/resting length) for varying degrees of stimulation and showed the similarity of myometrium to skeletal muscle. They included the resting (passive) state of myometrium as a limiting case, but they reported no stress values. Schofield and Wood (1964) used a similar approach to measure normalized tensile forces vs normalized length for active states of pregnant and nonpregnant rabbit and human myo’ Received 30 September 1977 Nocember 1977.
; receiced for publication 3
relating
resting
length
to in cico length
at
different stages of pregnancy. Wood measured the relationship between tensile force and length for excised strips of human myomet-
tium in both active (1964a) and passive (1964b) states, and compared the optimal length for muscular contraction in vitro with the in uico length of myometrium at various stages of pregnancy. He estimated crosssectional areas from sample weights and lengths and presented the mechanical behavior in the passive state in terms of tensile stress vs length. Loading was accomplished in 20gm increments each half-min until the tissue ruptured. From Wood’s data in the passive state, an approximate tensile strength can be calculated for myometrium at rupture (ca. 70 psi). which we infer is a true stress (F/A), rather than an engineering stress (F/A,), because the lengths from which areas were calculated probably were measured under load (Wood, personal communication). Wood also used dead-weight loading of 100 gm for 1 min to arbitrarily divide the deformation into ‘elastic’ and ‘plastic’ components, although both exhibited time-dependent behavior which he did not analyze explicitly. Conrad et al. (1966a, b, 1967) demonstrated the importance of time-dependent mechanical behavior of myometrium in stress-relaxation and step-loading tensile tests. According to their procedure for measuring passive tensile properties (Conrad, 1966b), crosssectional areas were measured optically; since these measurements were made before applying tension to the tissue (Conrad, personal communication), the stress values reported are engineering stress. The tensile stress vs length data they reported are instructive, even though some stress-relaxation was permitted between loading steps. The principal objective of the research reported in the present paper was to characterize the range of passive mechanical behavior of typical human myometrium, in both tension and compression, at modest strain rates.
167
G. W. PEARSALL
168
EXPERIMENTAL
and V. L. ROBERTS
PROCEDURES
Each intact uterus was excised during a normal vaginal hysterectomy performed at the Duke University Medical Center. The uteri usually were ‘butterflied’ by making a longitudinal incision from’the cervix to the fundus to permit opening the uterus for inspection and removal of specimens. Some uteri contained small cysts or fibroids; these portions were avoided in choosing regions for measuring mechanical ’ properties, Uteri with malignancies were not used in this study. All mechanical properties reported were measured on specimens from the upper uterine wall and fundus; the cervical region, oviduct attachments, and ligament insertions were avoided. Mechanical properties were measured as soon as possible following removal by the surgical team, always within 2 hr of excision for data reported here. After the tests were performed, all specimens were returned for routine pathological examination and disposal. Tension specimens were cut from the uterine tissue by making approximately parallel cuts, usually cu. 1/4-3/g in. apart, with a scalpel. A gauge section was then carved by scalpel, cu. 1 in. long with slightly concave sides (radius of curvature > 10 in.) and a minimum width between l/4 and l/2 in. To promote gripping without severe damage to the tissue, strips of thick rubber band were glued to the specimen ends (Fig. 1) with a cyanoacrylate adhesive. These rubberfaced specimen ends were then gripped between rubber-faced grips in an Instron Testing Machine for tension testing. Each sample was permitted to relax for cu. 5 min before testing. The gauge length e,, was taken as the distance between the rubber-band grip pads after relaxation. Tension was applied by moving the cross-head at 0.5 in./min, which resulted in tensile strain rates of cu. 0.2-0.5/min. Although preliminary tests confirmed that myometrium is viscoelastic, changing the strain rate by a factor of ten produced less than a two-fold variation in stress, so all tests were performed at approximately the same strain rate. Radial core compression specimens were taken by means of a specially sharpened cork-borer with an inner diameter of approx. 0.7 in. (slightly less than 23/32 in.). All cores were taken along a radial axis with respect to the approximately spherical geometry of the upper uterus (Fig. 2). Some relaxation occurred immediately after coring, usually to a uniform diameter of cc. l/2 in, but occasionally resulting in a transformation of the cored cylinder to a somewhat ‘lumpy geometry. When the core was approx. 1 in. or less long, the endometrium and exterior surface were trimmed from it to produce a cylinder approx. l/2-5/8 in. long. When the core was over 1 in. long, the cylinder was cut into two shorter cylinders after trimming the endometrium and the exterior surface; this latter procedure resulted in both a more stable geometry for compression testing (avoided problems of buckling) and also permitted a comparison of mechanical properties between inner and outer layers ofmyometrium.
Compression testing was performed in an Instron Testing Machine, using the compression cage assembly shown in Fig. 3. The cylinder ends were ‘lubricated’ by films of polypropylene placed between the specimen and the aluminium surfaces of the compression cage. Compression was accomplished normally at a cross-head rate of 0.2 in.;min, which resulted in compressive strain rates of ~a. 0.5-l/min.
DATA
REDUCTION
AND
ANALYSIS
Force-displacement curves were recorded for all tests, then converted to stress-strain data as follows. At any point on the force-displacement curve, the engineering (Lagrangian) stress was obtained by dividing the force by the original cross-sectional area of the specimen F
rJe=-.
A0
Engineering
strain Af se=-=(II
E -to fo
’
(2)
was calculated by dividing the displacement by the original specimen gauge length. Engineering stress and strain are useful concepts because of their simplicity and because of their relevance to design, but for understanding the intrinsic properties of a material. true stress and strain - defined in terms of the instantaneous geometry of the specimen - are preferable. True strain is defined as E=
e fo’ IDdP+L t,.
and therefore can be calculated strain by e=ln(s,+
I).
from engineering
(4)
True stress can be calculated from engineering stress quite easily if the specimen volume remains constant. Measurement of specimen dimensions, before and after testing, suggested that constant volume is a reasonable approximation for myometrium. True stress is defined as F CT=-.
A
Assuming constant volume, u = U,(&,+ 1).
(6)
All stress-strain data presented are true stress-true strain, calculated by equations (4) and (6). A significant experimental problem exists in determining where ‘zero’ is on a force-displacement curve of a material like myometrium, because the force-displacement curve appears to approach the displacement axis asymptotically at zero. For the test results reported here, zero was established as follows.
Fig. 1. Tension
specimens
of uterine
muscle
ready
for testin:
G. W. PEARSLL
and V. L.
ROBERTS
Fig. 2. Uterus with compression specimen removed.
i-
173
Passive mechanical properties of uterine muscle
After zeroing the load-cell electronics, calibrating the load cell, and taring out the weight of the grips or compression cage, the load selection switch was turned to one lb full scale (10 in. of chart), and the cross-head was moved manually until a slight positive load was detected (cu. 0.005 lb); the cross-head was then backed off just to zero load, the load selection switch was turned to the test range (usually 20 or 50 lb full scale), and the test was started immediately. The load cell was calibrated by dead-weight loading before each series of tests on any one uterus; the electronic circuit was zeroed and balanced before each individual test. EXPEtiIMENYAL
RESULTS
The tensile properties of myometrium, measured during this investigation, are presented in Figs. 4-6. Figure 4 is a composite plot of the tensile true stress-true strain behavior of passive uterine muscle, plotted on semilogarithmic axes. Figure 5 shows the composite stress-strain behavior of the myometrium samples which exhibited tensile strengths at failure above 160 psi, plotted on linear stress and strain axes, while Fig. 6 presents similar data for myometrium samples which exhibited tensile strengths at failure below 160 psi. Ultimate failure in tension occurred by propagation of tears to produce final rupture, the tears commonly initiating at imperfections produced during excision of the samples from the intact uterus or during cutting of the gauge section. The compressive properties of myometrium, measured in this investigation, are presented in Figs. 7 and 8. Figure 7 is a composite plot of compressive true
lRtX.SlWNFig. 5. Composite stress-strain curves for uterine muscle in tension. The data shown are for those specimens which failed
TRUEZJRAIN-’ Fig. 6. Composite stress-strain curves for uterine muscle in tension. The data shown are for those specimens which failed at stresses below approx. 160psi.
-+Fig. 4. Composite stress-strain curves for uterine muscle in tension, semilogarithmic plot.
stress-true strain behavior of passive uterine muscle, plotted on semilogarithmic axes. Figure 8 shows the composite stress-strain behavior of these same samples, plotted on linear axes.
G.
174
W. PEARSALL and V. L. ROBERTT 200-
loo-
M-
. I ?
a!
2
Kl-
S-
l-
a4
oh
0.8
-
Lo
TRUESTRAIN-
Fig. 7. Composite stress-strain curves for uterine muscle in compression, semilogarithmic plot.
Fig. 9. Tensile and compression data for three specimens of myometrium taken from a single uterus (history No. C47423). Curves A and B represent tensile data while curve C represents compression data.
a2
a4
a4
an
Lo
TRUESTRAIN--,
Fig. 8. Composite stress-strain curves for uterine muscle in compression.
a4
0.6
0.8
Lo
L2
The mechanical properties of three uteri, from which both tension and compression data could be obtained, are represented in Figs. 9-11 as true stress-true strain curves plotted on semilogarithmic axes.
Fig. 10. Tensile and compression data for three specimens of myometrium taken from a single uterus (history No. E92125). Curves A and B represent tensile data while curve C represents compression data.
DISCUSSION
strain for a given test can be approximated for stresses above 1 psi by the straight line
From the behavior of passive myometrium in both tension and compression, as plotted on semilogarithmic axes, the relationship between true stress and true
E=Clna+B.
(7)
Although stresses below 1 psi are of little interest in
Passive mechanical properties of uterine muscle
TRUEWN-
Fig. 11. Tensile and compression data for four specimens of myometrium taken from a single uterus (history No. K65450). This figure illustrates the consistency of tensile data (Curves A & B) and compression data (Curves C & D) from one uterus. of labor or of damage by penetration, the portion of the stress-strain curve below 1 psi could be represented by
studies
E = Ba
forjoj
Ii,
(8)
which satisfies theconditions that equations (7) and (8) are equivalent at 0 = 1, and stress equals zero when strain equals zero. That myometrium stress-strain curves in both tension and compression satisfy the linearized equation (7) approximately means that the resistance to additional deformation increases exponentially with strain. As Fung (1967.1970) has pointed out, e.xponential stress-strain relationships are typical of many biological materials; such behavior is similar to that of a spring that stiffens under load. The stiffness of tissue which obeys equation (7) can be described by a differential elastic modulus (sometimes called a tangent modulus)
&&?_a d&-C’
(9)
The constant l/C can be determined easily as the slope of the stress-strain curve, plotted semilogarithmically (In a) 1 = -. ds C At equal values of stress, the elastic stiffness of two specimens will be proportional to the slopes of their In (3 - E curves. Choosing 10 psi somewhat arbitrarily for comparing different uteri, almost all specimens
175
exhibited elastic moduli at that stress between 70 and 200 psi. Tensile strength values at failure ranged from CCI. 80-300 psi. the lower strengths usually associated with the propagation of minor cuts produced during specimen preparation. The failure mechanism of slow tearing invariably was associated with convexity of the In c - E curve prior to rupture. When the data of Conrad et al. on passive tensile properties of myometrium (1966b) are converted to true stress and true strain and are plotted on semilogarithmic coordinates, they fall within the range of behavior represented by Fig. 4. His curve for nonpregnant myometrium falls approximately in the middle of the curves in Fig. 4 and exhibits an elastic modulus, at an applied stress of lOpsi, equal to cu. 140 psi. His curve for pregnant myometrium overlays the curves at the right of Fig. 4 and exhibits an elastic modulus, at an applied stress of lOpsi, equal to ca. 85 psi. Three of the curves at the right of Figure 4 also represents pregnant myometrium. The stress relaxation which Conrad er al. permitted between their data points (1966b) presumably reduced the value of elastic modulus from that which would have been obtained in a continuous test. but the present comparison suggests that the differences maq be small, compared to statistical scatter over many samples. The comparison also tends to support our presumption that mechanical testing of myometrium in air is satisfactory, within the statistical scatter to be expected from such tissue, if the testing is performed soon after excision. When the data from Wood (1964b) on a myometrial strip from the lower uterine segment (“Caesarean section for foetal distress. Cervix 3-fingers dilated. Foetal weight : 4500 gm”) are plotted on semilogarithmic axes, the resulting curve also falls within the range of our data, exhibiting an elastic modulus, at a stress of 10 psi, equal to cu. 175 psi. The stiffness of myometrium (equations 9 and 10) in compression tends to be less than, or occasionallt about equal, to the stiffness in tension, measured at the same stress (Figs. 9-11). However, at equivalent strains, the stiffness of myometrium in compression is considerably less than in tension, as the compression stress-strain curve is always shifted to the right of the tension curve. The reason for this difference in stiffness probably lies in the nature of the collagen network in myometrium (Mosler, 1968); the collagen network can be stretched taut in tension, but not in compression. When a cored compression sample was divided to produce specimens of both inner and outer tissue, the outer tissue invariably was less stiff, or more compliant. By comparison, the outer strip in tension testing invariably was stiffer than the inner strip. We speculate that this difference is related to the anisotropy of fiber orientation in the collagen network, or possibly even to muscle-fiber orientation (Mizrahi, 1973). In summary, we have determined the tensile and compressive behavior of human myometrium at mo-
G. W. F%ARSALLand V.L. ROBERTS
176
derate strain rates. The results achieved agree with the limited data which had previously appeared in the literature and illustrate the considerable variation to be expected among different samples. Uterine muscle exhibits stress-strain behavior similar to that of a stiffening spring, a characteristic which is typical of soft tissue. Acknowledgements - This research was made possible by a
grant from grateful to mechanical the clinical research.
the A. H. Robins Company. The authors are Dr. A. Mertl for assistance in determining the properties and to Dr. R. G. Brame for arranging assistance so necessary to the pursuit of this
REFERENCES Bell, F. (1972) Biomechanics of human parturition: a fundamental approach to the first stage of labour. Ph.D. thesis, Bioengineering Unit, University of Strathclyde, Glasgow, Scotland. Csapo, A. I. and Goodall. M. (1954) Excitability, length tension relation and kinetics of uterine muscle contraction in relation to hormonal status. J. Physiol.. Lond. 126, 348-395. Conrad, J. T., Kuhn, W. K. and Johnson, W. L. (1966a) Stress relaxation in human uterine muscle. Am. J. Obsret. Gynec. 95, 254-265.
Conrad, J. T., Johnson, W. L., Kuhn, W. K. and Hunter, C. A. (1966b) Passive stretch relationships in human uterine muscle. Am. J. Obsret. Gynec. %, 1055-1059. Conrad. J. T. and Kuhn. W. K. (1967) The active length-tension relationship in human uterine muscle, Am. J. Obster. Gynec. 97, 154-160. Conrad, J. T.. personal communication. Fung, Y. C. (1967) Elasticity of soft tissues in simple elongation. &I. J. Pfrysiol. 213, 1532-1544. Fung. Y. C. (1970) Mathematical representation of the mechanical properties of the heart muscle. J. Biomechnnics 3, 381-404. Mizrahi, J. (1975) Deformation analysis of the uterine muscle during labor and delivery. The Julius Silver Institute of Biomedical Engineering Sciences, Technion-Israel Institute of Technology, June. Mizrahi. J. and Kami, Z. (1975) A mechanical model for uterine muscle activity during labor and delivery. Israel J. Tech&. 13, 185-191. Mosler, K. H. (1968) The dynamics of uterine muscle, in Bibliorheca Gynaecologica, No. 48, Adcances in Obsrerrics and Gynaecology, Vol. 35. Karger, Basel.
Schofield, B. M. and Wood, C. (1964) Length-tension relation in rabbit and human myometrium. J. Physiol. 175, 125-133. Wilson. J. F., Tsui, Y. and Roberts, V. L. (1977) Plane penetration of uterine muscle by intrauterine shields (to be published). Wood, C. (1964a) Physiology of uterine contractions. J. Obsrer. Gynaec. Brir. Cwlfh. 71, 360-373.
Wood, C. (1964b) The expansile behavior of the human uterus. J. Obsrer. Gynaec. Brit. Cwlth. 71, 615-620. Wood, C., personal communication.
and V. L. ROBERTS
PFARSALL
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27706, U.S.A. Abstract-The mechanical properties of human myometrium in uirro were determined in the passive state at strain rates ofapprox. 0.5 min. Both tension and compression data are presented as curves of true stress vs true strain. The test results represent a range of uterine conditions, from nulliparous to multiparous, in various stages of the menstrual cycle, and include one pregnant uterus. For stresses above 1psi, either tension or compression data can be represented approximately by an equation of the form E = C In u + B. Elastic moduli, measured at an applied stress of 10 psi, were within the range 70-200 psi. and tensile strengths were in _ the range 8&300 psi.
INTRODUCTION
metrium,
The mechanical properties of uterine muscle (myometrium) are of fundamental importance for studying uterine activity during pregnancy and labor, as well as for assessing the susceptibility of the uterus to damage by penetration (Wilson et al., 1977). Yet relatively little stress-strain information has been published describing the mechanical behavior of this muscle, and a number of questions and contradictions exist among the published results. An extensive bibliography has been compiled by Bell (1972), summarizing the mechanical-property studies of myometrium which have been published. Tensile mechanical properties of strips excised from intact uteri have been measured by Csapo and Goodall (1954). Schofield and Wood (1964), Wood (1964a, b), Conrad et al. (1966a, 1966b, 1967), and Mosler (1968). To our knowledge, no compressive stressstrain measurements have been reported for myometrium. Also we are unaware of tensile stress-strain data having been measured from myometrium under conditions of approximately constant strain rate. In the first studies of length-tension relationships in uterine muscle, Csapo and Goodall (1954) excised strips of rabbit uteri and suspended them in saturated Krebs solution by platinum hooks attached to a tensometer, measuring tensile forces produced by electrical stimulation. They plotted normalized force (force/maximum force) vs normalized Length (length/resting length) for varying degrees of stimulation and showed the similarity of myometrium to skeletal muscle. They included the resting (passive) state of myometrium as a limiting case, but they reported no stress values. Schofield and Wood (1964) used a similar approach to measure normalized tensile forces vs normalized length for active states of pregnant and nonpregnant rabbit and human myo’ Received 30 September 1977 Nocember 1977.
; receiced for publication 3
relating
resting
length
to in cico length
at
different stages of pregnancy. Wood measured the relationship between tensile force and length for excised strips of human myomet-
tium in both active (1964a) and passive (1964b) states, and compared the optimal length for muscular contraction in vitro with the in uico length of myometrium at various stages of pregnancy. He estimated crosssectional areas from sample weights and lengths and presented the mechanical behavior in the passive state in terms of tensile stress vs length. Loading was accomplished in 20gm increments each half-min until the tissue ruptured. From Wood’s data in the passive state, an approximate tensile strength can be calculated for myometrium at rupture (ca. 70 psi). which we infer is a true stress (F/A), rather than an engineering stress (F/A,), because the lengths from which areas were calculated probably were measured under load (Wood, personal communication). Wood also used dead-weight loading of 100 gm for 1 min to arbitrarily divide the deformation into ‘elastic’ and ‘plastic’ components, although both exhibited time-dependent behavior which he did not analyze explicitly. Conrad et al. (1966a, b, 1967) demonstrated the importance of time-dependent mechanical behavior of myometrium in stress-relaxation and step-loading tensile tests. According to their procedure for measuring passive tensile properties (Conrad, 1966b), crosssectional areas were measured optically; since these measurements were made before applying tension to the tissue (Conrad, personal communication), the stress values reported are engineering stress. The tensile stress vs length data they reported are instructive, even though some stress-relaxation was permitted between loading steps. The principal objective of the research reported in the present paper was to characterize the range of passive mechanical behavior of typical human myometrium, in both tension and compression, at modest strain rates.
167
G. W. PEARSALL
168
EXPERIMENTAL
and V. L. ROBERTS
PROCEDURES
Each intact uterus was excised during a normal vaginal hysterectomy performed at the Duke University Medical Center. The uteri usually were ‘butterflied’ by making a longitudinal incision from’the cervix to the fundus to permit opening the uterus for inspection and removal of specimens. Some uteri contained small cysts or fibroids; these portions were avoided in choosing regions for measuring mechanical ’ properties, Uteri with malignancies were not used in this study. All mechanical properties reported were measured on specimens from the upper uterine wall and fundus; the cervical region, oviduct attachments, and ligament insertions were avoided. Mechanical properties were measured as soon as possible following removal by the surgical team, always within 2 hr of excision for data reported here. After the tests were performed, all specimens were returned for routine pathological examination and disposal. Tension specimens were cut from the uterine tissue by making approximately parallel cuts, usually cu. 1/4-3/g in. apart, with a scalpel. A gauge section was then carved by scalpel, cu. 1 in. long with slightly concave sides (radius of curvature > 10 in.) and a minimum width between l/4 and l/2 in. To promote gripping without severe damage to the tissue, strips of thick rubber band were glued to the specimen ends (Fig. 1) with a cyanoacrylate adhesive. These rubberfaced specimen ends were then gripped between rubber-faced grips in an Instron Testing Machine for tension testing. Each sample was permitted to relax for cu. 5 min before testing. The gauge length e,, was taken as the distance between the rubber-band grip pads after relaxation. Tension was applied by moving the cross-head at 0.5 in./min, which resulted in tensile strain rates of cu. 0.2-0.5/min. Although preliminary tests confirmed that myometrium is viscoelastic, changing the strain rate by a factor of ten produced less than a two-fold variation in stress, so all tests were performed at approximately the same strain rate. Radial core compression specimens were taken by means of a specially sharpened cork-borer with an inner diameter of approx. 0.7 in. (slightly less than 23/32 in.). All cores were taken along a radial axis with respect to the approximately spherical geometry of the upper uterus (Fig. 2). Some relaxation occurred immediately after coring, usually to a uniform diameter of cc. l/2 in, but occasionally resulting in a transformation of the cored cylinder to a somewhat ‘lumpy geometry. When the core was approx. 1 in. or less long, the endometrium and exterior surface were trimmed from it to produce a cylinder approx. l/2-5/8 in. long. When the core was over 1 in. long, the cylinder was cut into two shorter cylinders after trimming the endometrium and the exterior surface; this latter procedure resulted in both a more stable geometry for compression testing (avoided problems of buckling) and also permitted a comparison of mechanical properties between inner and outer layers ofmyometrium.
Compression testing was performed in an Instron Testing Machine, using the compression cage assembly shown in Fig. 3. The cylinder ends were ‘lubricated’ by films of polypropylene placed between the specimen and the aluminium surfaces of the compression cage. Compression was accomplished normally at a cross-head rate of 0.2 in.;min, which resulted in compressive strain rates of ~a. 0.5-l/min.
DATA
REDUCTION
AND
ANALYSIS
Force-displacement curves were recorded for all tests, then converted to stress-strain data as follows. At any point on the force-displacement curve, the engineering (Lagrangian) stress was obtained by dividing the force by the original cross-sectional area of the specimen F
rJe=-.
A0
Engineering
strain Af se=-=(II
E -to fo
’
(2)
was calculated by dividing the displacement by the original specimen gauge length. Engineering stress and strain are useful concepts because of their simplicity and because of their relevance to design, but for understanding the intrinsic properties of a material. true stress and strain - defined in terms of the instantaneous geometry of the specimen - are preferable. True strain is defined as E=
e fo’ IDdP+L t,.
and therefore can be calculated strain by e=ln(s,+
I).
from engineering
(4)
True stress can be calculated from engineering stress quite easily if the specimen volume remains constant. Measurement of specimen dimensions, before and after testing, suggested that constant volume is a reasonable approximation for myometrium. True stress is defined as F CT=-.
A
Assuming constant volume, u = U,(&,+ 1).
(6)
All stress-strain data presented are true stress-true strain, calculated by equations (4) and (6). A significant experimental problem exists in determining where ‘zero’ is on a force-displacement curve of a material like myometrium, because the force-displacement curve appears to approach the displacement axis asymptotically at zero. For the test results reported here, zero was established as follows.
Fig. 1. Tension
specimens
of uterine
muscle
ready
for testin:
G. W. PEARSLL
and V. L.
ROBERTS
Fig. 2. Uterus with compression specimen removed.
i-
173
Passive mechanical properties of uterine muscle
After zeroing the load-cell electronics, calibrating the load cell, and taring out the weight of the grips or compression cage, the load selection switch was turned to one lb full scale (10 in. of chart), and the cross-head was moved manually until a slight positive load was detected (cu. 0.005 lb); the cross-head was then backed off just to zero load, the load selection switch was turned to the test range (usually 20 or 50 lb full scale), and the test was started immediately. The load cell was calibrated by dead-weight loading before each series of tests on any one uterus; the electronic circuit was zeroed and balanced before each individual test. EXPEtiIMENYAL
RESULTS
The tensile properties of myometrium, measured during this investigation, are presented in Figs. 4-6. Figure 4 is a composite plot of the tensile true stress-true strain behavior of passive uterine muscle, plotted on semilogarithmic axes. Figure 5 shows the composite stress-strain behavior of the myometrium samples which exhibited tensile strengths at failure above 160 psi, plotted on linear stress and strain axes, while Fig. 6 presents similar data for myometrium samples which exhibited tensile strengths at failure below 160 psi. Ultimate failure in tension occurred by propagation of tears to produce final rupture, the tears commonly initiating at imperfections produced during excision of the samples from the intact uterus or during cutting of the gauge section. The compressive properties of myometrium, measured in this investigation, are presented in Figs. 7 and 8. Figure 7 is a composite plot of compressive true
lRtX.SlWNFig. 5. Composite stress-strain curves for uterine muscle in tension. The data shown are for those specimens which failed
TRUEZJRAIN-’ Fig. 6. Composite stress-strain curves for uterine muscle in tension. The data shown are for those specimens which failed at stresses below approx. 160psi.
-+Fig. 4. Composite stress-strain curves for uterine muscle in tension, semilogarithmic plot.
stress-true strain behavior of passive uterine muscle, plotted on semilogarithmic axes. Figure 8 shows the composite stress-strain behavior of these same samples, plotted on linear axes.
G.
174
W. PEARSALL and V. L. ROBERTT 200-
loo-
M-
. I ?
a!
2
Kl-
S-
l-
a4
oh
0.8
-
Lo
TRUESTRAIN-
Fig. 7. Composite stress-strain curves for uterine muscle in compression, semilogarithmic plot.
Fig. 9. Tensile and compression data for three specimens of myometrium taken from a single uterus (history No. C47423). Curves A and B represent tensile data while curve C represents compression data.
a2
a4
a4
an
Lo
TRUESTRAIN--,
Fig. 8. Composite stress-strain curves for uterine muscle in compression.
a4
0.6
0.8
Lo
L2
The mechanical properties of three uteri, from which both tension and compression data could be obtained, are represented in Figs. 9-11 as true stress-true strain curves plotted on semilogarithmic axes.
Fig. 10. Tensile and compression data for three specimens of myometrium taken from a single uterus (history No. E92125). Curves A and B represent tensile data while curve C represents compression data.
DISCUSSION
strain for a given test can be approximated for stresses above 1 psi by the straight line
From the behavior of passive myometrium in both tension and compression, as plotted on semilogarithmic axes, the relationship between true stress and true
E=Clna+B.
(7)
Although stresses below 1 psi are of little interest in
Passive mechanical properties of uterine muscle
TRUEWN-
Fig. 11. Tensile and compression data for four specimens of myometrium taken from a single uterus (history No. K65450). This figure illustrates the consistency of tensile data (Curves A & B) and compression data (Curves C & D) from one uterus. of labor or of damage by penetration, the portion of the stress-strain curve below 1 psi could be represented by
studies
E = Ba
forjoj
Ii,
(8)
which satisfies theconditions that equations (7) and (8) are equivalent at 0 = 1, and stress equals zero when strain equals zero. That myometrium stress-strain curves in both tension and compression satisfy the linearized equation (7) approximately means that the resistance to additional deformation increases exponentially with strain. As Fung (1967.1970) has pointed out, e.xponential stress-strain relationships are typical of many biological materials; such behavior is similar to that of a spring that stiffens under load. The stiffness of tissue which obeys equation (7) can be described by a differential elastic modulus (sometimes called a tangent modulus)
&&?_a d&-C’
(9)
The constant l/C can be determined easily as the slope of the stress-strain curve, plotted semilogarithmically (In a) 1 = -. ds C At equal values of stress, the elastic stiffness of two specimens will be proportional to the slopes of their In (3 - E curves. Choosing 10 psi somewhat arbitrarily for comparing different uteri, almost all specimens
175
exhibited elastic moduli at that stress between 70 and 200 psi. Tensile strength values at failure ranged from CCI. 80-300 psi. the lower strengths usually associated with the propagation of minor cuts produced during specimen preparation. The failure mechanism of slow tearing invariably was associated with convexity of the In c - E curve prior to rupture. When the data of Conrad et al. on passive tensile properties of myometrium (1966b) are converted to true stress and true strain and are plotted on semilogarithmic coordinates, they fall within the range of behavior represented by Fig. 4. His curve for nonpregnant myometrium falls approximately in the middle of the curves in Fig. 4 and exhibits an elastic modulus, at an applied stress of lOpsi, equal to cu. 140 psi. His curve for pregnant myometrium overlays the curves at the right of Fig. 4 and exhibits an elastic modulus, at an applied stress of lOpsi, equal to ca. 85 psi. Three of the curves at the right of Figure 4 also represents pregnant myometrium. The stress relaxation which Conrad er al. permitted between their data points (1966b) presumably reduced the value of elastic modulus from that which would have been obtained in a continuous test. but the present comparison suggests that the differences maq be small, compared to statistical scatter over many samples. The comparison also tends to support our presumption that mechanical testing of myometrium in air is satisfactory, within the statistical scatter to be expected from such tissue, if the testing is performed soon after excision. When the data from Wood (1964b) on a myometrial strip from the lower uterine segment (“Caesarean section for foetal distress. Cervix 3-fingers dilated. Foetal weight : 4500 gm”) are plotted on semilogarithmic axes, the resulting curve also falls within the range of our data, exhibiting an elastic modulus, at a stress of 10 psi, equal to cu. 175 psi. The stiffness of myometrium (equations 9 and 10) in compression tends to be less than, or occasionallt about equal, to the stiffness in tension, measured at the same stress (Figs. 9-11). However, at equivalent strains, the stiffness of myometrium in compression is considerably less than in tension, as the compression stress-strain curve is always shifted to the right of the tension curve. The reason for this difference in stiffness probably lies in the nature of the collagen network in myometrium (Mosler, 1968); the collagen network can be stretched taut in tension, but not in compression. When a cored compression sample was divided to produce specimens of both inner and outer tissue, the outer tissue invariably was less stiff, or more compliant. By comparison, the outer strip in tension testing invariably was stiffer than the inner strip. We speculate that this difference is related to the anisotropy of fiber orientation in the collagen network, or possibly even to muscle-fiber orientation (Mizrahi, 1973). In summary, we have determined the tensile and compressive behavior of human myometrium at mo-
G. W. F%ARSALLand V.L. ROBERTS
176
derate strain rates. The results achieved agree with the limited data which had previously appeared in the literature and illustrate the considerable variation to be expected among different samples. Uterine muscle exhibits stress-strain behavior similar to that of a stiffening spring, a characteristic which is typical of soft tissue. Acknowledgements - This research was made possible by a
grant from grateful to mechanical the clinical research.
the A. H. Robins Company. The authors are Dr. A. Mertl for assistance in determining the properties and to Dr. R. G. Brame for arranging assistance so necessary to the pursuit of this
REFERENCES Bell, F. (1972) Biomechanics of human parturition: a fundamental approach to the first stage of labour. Ph.D. thesis, Bioengineering Unit, University of Strathclyde, Glasgow, Scotland. Csapo, A. I. and Goodall. M. (1954) Excitability, length tension relation and kinetics of uterine muscle contraction in relation to hormonal status. J. Physiol.. Lond. 126, 348-395. Conrad, J. T., Kuhn, W. K. and Johnson, W. L. (1966a) Stress relaxation in human uterine muscle. Am. J. Obsret. Gynec. 95, 254-265.
Conrad, J. T., Johnson, W. L., Kuhn, W. K. and Hunter, C. A. (1966b) Passive stretch relationships in human uterine muscle. Am. J. Obsret. Gynec. %, 1055-1059. Conrad. J. T. and Kuhn. W. K. (1967) The active length-tension relationship in human uterine muscle, Am. J. Obster. Gynec. 97, 154-160. Conrad, J. T.. personal communication. Fung, Y. C. (1967) Elasticity of soft tissues in simple elongation. &I. J. Pfrysiol. 213, 1532-1544. Fung. Y. C. (1970) Mathematical representation of the mechanical properties of the heart muscle. J. Biomechnnics 3, 381-404. Mizrahi, J. (1975) Deformation analysis of the uterine muscle during labor and delivery. The Julius Silver Institute of Biomedical Engineering Sciences, Technion-Israel Institute of Technology, June. Mizrahi. J. and Kami, Z. (1975) A mechanical model for uterine muscle activity during labor and delivery. Israel J. Tech&. 13, 185-191. Mosler, K. H. (1968) The dynamics of uterine muscle, in Bibliorheca Gynaecologica, No. 48, Adcances in Obsrerrics and Gynaecology, Vol. 35. Karger, Basel.
Schofield, B. M. and Wood, C. (1964) Length-tension relation in rabbit and human myometrium. J. Physiol. 175, 125-133. Wilson. J. F., Tsui, Y. and Roberts, V. L. (1977) Plane penetration of uterine muscle by intrauterine shields (to be published). Wood, C. (1964a) Physiology of uterine contractions. J. Obsrer. Gynaec. Brir. Cwlfh. 71, 360-373.
Wood, C. (1964b) The expansile behavior of the human uterus. J. Obsrer. Gynaec. Brit. Cwlth. 71, 615-620. Wood, C., personal communication.