Determination of Elastic Parameters For Human Fetal Membranes

Determination of Elastic Parameters For Human Fetal MembranesC. E. Miller, J. P. Lavery, and T. A. Donnelly Citation: J. Rheol. 23, 57 (1979); doi: 10.1122/1.549534 View online: http://dx.doi.org/10.1122/1.549534 View Table of Contents: http://www.journalofrheology.org/resource/1/JORHD2/v23/i1 Published by the The Society of Rheology Related ArticlesLigament creep behavior can be predicted from stress relaxation by incorporating fiber recruitment J. Rheol. 45, 493 (2001) The Mechanical Sensitivity of Soft Compressible Testing Machines J. Rheol. 33, 455 (1989) Oscillatory Viscometry of Red Blood Cell Suspensions: Relations to Cellular Viscoelastic Properties J. Rheol. 30, 231 (1986) Use of a Rheological Technique to Evaluate Erythrocyte Membrane Alterations J. Rheol. 23, 721 (1979) Surface Elasticity and Viscosity of Red Cell Membrane J. Rheol. 23, 669 (1979) Additional information on J. Rheol.Journal Homepage: http://www.journalofrheology.org/ Journal Information: http://www.journalofrheology.org/about Top downloads: http://www.journalofrheology.org/most_downloaded Information for Authors: http://www.journalofrheology.org/author_information

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Determination of Elastic Parameters For Human

Fetal Membranes

C. E. MILLER, Speed Scientific School, University of Louisville,Louisville, Kentucky; J. P. LAVERY, School of Medicine,University of Louisville, Louisville, Kentucky; and T. A.DONNELLY,Owens Construction Company, Louisville,

Kentucky

Synopsis

In the present research circular sheets of rubber or soft fetal membrane tissue areinflated within the confines of a cylindrical tube under hydrostaiic pressure. The resulting deformations, which are surfaces of revolution, are described and analyzed.The experimental method of deforming the membranes was chosen so that the deformation profiles would have physiological similarity to the expansions of the chorioamniotic sac during the period of labor in the final stage of pregnancy. Thekinematical response of the fetal membranes is shown to be the same as the kinematicalresponse of rubber. The principal extension ratios are calculated for a rubber membrane for several deformation profiles. From data obtained in deflection experimentswith rubber an approximate deflection profile is established. Assuming rubber to bea Mooney material, and using the theory of elastic membranes described by Green andAdkins, displacements, slopes, and second derivatives of a refined deformation profilefor rubber are derived as a function of the maximum displacement at the pole. Usingthese results it is shown that one can evaluate the elastic parameters of human fetalmembranes. In this manner one can make a meaningful comparison of the elasticnature of fetal membranes from term pregnancies to the elastic nature of membraneswhich rupture at various gestational stages.

INTRODUCTION

Premature rupture of the chorioamniotic sac occurs quite frequently, and the cause of the rupture is not adequately known.' Therange of incidence of premature rupture is between 7 and 12%of thetotal number of births and it is nowrecognized that a greater incidenceof abnormalities exist in babies born prematurely than among thoseborn in a term pregnancy.

A review of past research carried out to determine the strength

© 1979 by The Society of Rheology, Inc. Published by John Wiley & Sons, Inc.Journal of Rheology, 23(1), 57-78 (1979) 0148-6055/79/0023-0057$01.00


58 MILLER, LAVERY, AND DONNELLY

characteristics of fetal membranes can be found in reports by Artaland Danforth.s Several other insights into the mechanical responseof the membrane to applied loads can be gained from the reports ofMacl.aughlin," Polishuk," and Parry-Jones." Recently the presentprincipal authors demonstrated the viscoelastic nature of the humanchorioamniotic membrane," However, the literature does not containattempts to develop a strain energy function for the human fetalmembrane nor to establish elastic parameters rooted in the theoryof finite elasticity.

In the present research either rubber or fetal membrane materialare placed in a biaxial stress state under hydrostatic pressure resultingin a finite deformation of the membranes. The kinematic responseof the membranes is described. The deformation profiles of therubber are analyzed using the theory of elastic membranes developedby Green and Adkins.f The results indicate that assuming rubberand fetal membranes to have the strain energy function developedby Rivlin? for a Mooney material, one is able to develop meaningfulelastic parameters for human fetal membranes.

EXPERIMENTAL DEFORMATION FUNCTIONS

Deflection Functions

The apparatus in which the membrane samples were inflated underhydrostatic pressure is described in a paper concerning the viscoelasticnature of fetal membranes.? and a recent Master's thesis.!" In theexperiments, a square sample of "gum" rubber or chorioamnioticmembrane material 15 em X 15 em is placed in a specimen holder. Asnug-fitting O-ring is placed around the outer diameter of the specimen holder, over the membrane, stretching the membrane smoothlyacross the 7.62-cm-diameter hole machined in the specimen holder.The sample thus prepared is then positioned onto the flange of a cylindrical tube (the lower chamber in Fig. 1). Four holes drilled in thespecimen holder are aligned with four bolts permanently mountedin the flange of the lower chamber. Two O-rings embedded in theflat surface of the specimen holder interlock with two mating circulargrooves, cut in the flange, pressing the membrane into the grooves,and clamping the membrane between the O-rings and the basementof the grooves. Prior to positioning the membrane holder in the apparatus, the lower chamber is filled with pseudoamniotic fluid-! whenthe membrane being examined is chorioamniotic material or filled


HUMAN FETAL MEMBRANES 59

MINUMFLECTIONODS

TIONARYFERENCE00I -STA

RE

13cm R-l-~ALU

DER

IiI I

II I ,

:II I I I I II I , I I I I

I I

DEFLECTION GAGEIN UNDEFLECTED

POSITION

LOWERCHAMBER

IMEMBRANE

-7.62em-

- SPECIMENHOLDER AND

FLANGE

io crnLIQUID

VALVE

~

PRESSURETAP===t=l

t==l:J;:::"'- EXHAUSTLINE

----17em-----oj

Fig. 1. Deflection gauge attachment and apparatus used to study deflection ofmembranes under hydrostatic pressure.

with water when the membrane studied is "gum" rubber. While theclamping ofthe membrane is being completed, care is taken to eliminate any air gap, providing only a liquid-membrane interface.

A deflection gauge, consisting of thin aluminum rods (shown in Fig.1), is mounted on the bolts protruding above the membrane holder.Wing nuts are then tightened down on the bolts clamping all units



solidly together. Using a specially designed ink stamp, a polarcoordinate system is printed onto the membrane prior to attachingthe deflection gauge. A flexible tube attached to the exhaust line isled to a graduated cylinder. The line is completely filled with liquidso that no air space exists in the tube. On the other side of thechamber a pressure line leads to a transducer and a physiograph whichcan record negative pressures. The entire unit in Fig. 1 is mountedon a motor-driven elevator platform which can be raised to specifiedheights above the graduated cylinder. The apparatus and instrumentation are shown in Fig. 2.

When the elevator platform is elevated to a specified height abovethe graduated cylinder, the valve on the exhaust tube is opened andliquid flows into the graduated cylinder until equilibrium is reached.The membrane deflects as a surface of revolution into the lowerchamber, with the liquid always remaining in contact with themembrane surface. As the membrane deflects, the aluminum rodsdrop with the membrane, describing the deflection surface (see Fig.1). The movable rods are placed at the pole and at three radial positions on opposite sides of the pole. These points have nondimensional radial measures of 0.25,0.50, and 0.75. Two reference rods,which remain stationary during deflection, are placed at the fixedboundary. The deflection of the rods is measured using a microscopemounted on a micrometer slide held in a horizontal position. Theslide has a full-scale reading of 25 mm and an accuracy of 0.005 mm.The vacuum created by the fluid evacuation is measured on thephysiograph.

As the membrane deflects, the circles on the polar coordinate system increase in radius. The radius can be measured using the microscope on the micrometer slide, held in a vertical position.

Experimental data are obtained in this manner for six specimensof "gum" rubber and for six specimens of fetal membrane taken fromterm pregnancies. The relation of the nondimensional vertical deflections along the radius as a function of the nondimensional maximum vertical deflection for rubber membranes is shown in Fig. 3.The solid lines are the best straight line fit of the data shown, whichare the average values of deflection at each radial point for the sixspecimens used. The lines were determined using a least-squares fittechnique of data points by successive polynomials and examinationof the standard deviation about the regression curve in each case. Theslope, intercept, and standard deviation of these lines are given inTable 1. The procedure was repeated for the fetal membranes. Theresults are given in Fig. 4 and Table 1.


HUMAN FETAL MEMBRANES

Fig. 2. Photograph of the deflection apparatus and monitoring equipment.

61



1.0f',

0.9

O.B

0.7

0.6

W/wO

0.5

0.4

0.3 f', - X = 0.25

.- X = 0.500.2 0- X = 0.75

0.1

00 0,/ 0.2 0.3 0.4 05 0.6 0.7

Wo/a

Fig. 3. Relation of the average nondimensional vertical deflections along the radiusas a function of the nondimensional maximum deflection for rubber membranes. wis the deflection at a nondimensional radial distance x = pia. Wo = maximum deflection at the pole. Radius a = 38.1 mm.

In a cross section the deflection profiles are a family of curves withzero deflection at the boundary and a maximum deflection at the pole.Using cylindrical coordinates a conceivable representation of theseprofiles in nondimensional coordinates for 1> = constant plane is

W' = (1- x 2) (1 + nf,= b2n+2x2n+2),n=l

(1)

which is suggested by the boundary conditions and the axisymmetricnature of the deflected surface. The nondimensional coordinates anddeflections are

px= ,

a1> = 1>,

Zz= ,

ZoW'=~

wo'(2)

where a is the radius of the undeformed membrane and p is the radialdistance of a point on the deformed membrane measured from the



TABLE!

Results of Curve-Fitting Analysis for Membranes Deflected in the ApparatusShown in Figure 1

63

Line At Slope InterceptStandardDeviation

x = 0.25X = 0.50X = 0.75

X = 0.25X = 0.50X ~ 0.75

X = 0.25X = 0.50X = 0.75

Rubber Membranes0.02 0.950.16 0.740.33 0.42

Fetal Membranes0.07 0.920.18 0.750,39 0.41

Theoretical Analysis0.02 0.940.18 0.750.30 0.44

0.010.020.02

0.010.020.03

pole. Also, w is the deflection at this radial distance and Wo is themaximum deflection at the pole.

The experimental results shown in Figs. 3 and 4 indicate that

Wow' = k-+B,

a(3)

where k and B are functions of x measured outward from the pole.The intercepts of the three straight lines shown in Figs. 3 and 4 areapproximately equal to the value of (l - x2 ) in each case. If onechooses a truncated form of Eq. (1) such that

w' = (1 - x 2)(1 + A 1x 4 + A2x 6 ) , (4)

where Al = -A z and Al = nwo/a, one observes that

Wow' == nx 4(1 - x Z)2 - + (1- X Z),

a(5)

which has the same form as Eq. (3) and satisfies the boundary condition of zero deflection at x = 1. When n = 5 one obtains

and

B = (1- X Z).

(6)

(7)



1.0

0.9 s

0.8

0.7

0.6

W/w 0.50

0.4

0.3 f, - X = 0.25

0.2• - X = 0.50

o - X = 0.75

0.1

O-l---.-----.-.-----.---r--.,----,a 0.1 0.2 0.3 0.4 0.5 0.6 0.7

WO/o

Fig. 4. Relation of the nondimensional vertical deflections along the radius as afunction of the nondimensional maximum deflection for fetal membranes. w is thedeflection at a nondimensional radial distance x = pia. Wo =maximum deflectionat the pole. Radius a = 38.1 mm.

(9)

(8)

(10)

The slopes and intercepts calculated from Eqs, (6) and (7) are shownin comparison to the experimental results in Table 1. Profiles developed from the calculations using Eq. (5) with n = 5 are show in Fig.7 (dotted lines) along with the average deflection points found in theexperiments.

Using Eq. (5) as an approximate function of the deflection profilesthe first and second derivatives can be obtained:

dw Wo- = -2x + (20x 3 - 60x 5 + 40x 7) -dx a

d 2w' Wo-- = -2 + (60x 2 - 300x 4 + 280x 6 ) - •dx 2 a

These values are only approximate. However, when they are accurately known, the longitudinal and latitudinal curvatures can beobtained. One can refer to these as K 1 and K 2, respectively:

Wo (d2W') [ w5 (dW') 2] -3/2K 1 = - -- 1+--a 2 dx? a 2 dx



Wo (dW') [ W5 (dW')2]-1/2K =- - 1+--2 a2x dx a 2 dx .

Extension Ratio Functions

65

(11)

As the membranes are deflected, the radii of several circles circumscribed on the membrane surface are measured. The enlargedradius is called p, corresponding to a deflection W in the deformedstate. The point on the deformed membrane whose coordinates are(p,w) is originally at the point (r,0) in the undeformed state. In theremainder of the paper, Xi and Xj will represent the nondimensionalradial distance from the pole in the undeformed and the deformedstate, respectively. Hence, for a membrane of radius a,

rXi =

aand (12)

Using the average values of six rubber specimens and six fetalmembrane specimens, the logarithms of plr are plotted against thenon dimensional vertical deflection at the pole (wola), for severaldeformation states at Xi = 0.15,0.25,0.35, and 0.45. The results areshown in Fig. 6. Since plr is the latitudinal extension ratio '>"2, theresults shown in Fig. 5 suggest that

A2 = Ao exp(m) wola, (13)

where Ao, the initial stretch ratio when Wo = 0, is equal to unity. InEq. (13) the value of m is the slope of the lines shown in Fig. 5, and mis a function of Xi. An algebraic expression for m which representsall curves simultaneously is

m = [x;(1 - X[)2 + 0.9(1 - X[)6] (14)

and Eq. (13) becomes for Ao = 1,

A2 = exp{ [xr(l- xr)2 + 0.9(1- X[)6] :o}. (15)

Yang and Feng12 have shown that the longitudinal stretch ratio 1..1can be obtained as the ratio of the deformed meridian arc length tothe undeformed meridian arc length. Therefore,

(16)

Since


66

.6

.5

MILLER, LAVERY, AND DONNELLY

A

oB

.4

~.3

.2

.1 .2

o 0

.3

o

.4 .5

WO/a

o

.6 .7

c

.8

D

.9

(17)

Fig. 5. Experimental latitudinal stretch ratios as a function of the nondimensionalmaximum vertical deflection for rubber and fetal membranes. Circled points are forrubber, squares are for fetal membranes. For curve A points are at Xi = 0.15; curveB, Xi = 0.25; curve C, Xi = 0.35; curve D, Xi = 0.45.

dw dui d p

d; = dp dr '

it follows that

Further,

and

dp [ (dW) 2]1/2Al =- 1 + - .dr dp

(18)

(19)

dp = A + r dAz (20)dr 2 dr '

where A2 is a decreasing function of r and differentiation is carriedout with respect to a decreasing r. Putting Eq. (20) into Eq, (18) andusing the relation developed in Eq. (15),

Al = A2 {I + [ 4xtO - xf) - 2xf(1 - xlP

+ 1O.8x;(1 - X;)5] :O}[1 + :5 (~:;rr/2. (21)



3.0

20

Eu

1.0

50 6010OL-_-'-_----'-__L-_-'-_-L_--'

o 20 30 40P (mmHg)

Fig. 6. Deflection at the pole (Wo) of the rubber membrane as a function of theapplied pressure (P).

If one assumes that both rubber and fetal membranes are incompressible, then the extension ratio A3 perpendicular to the plane ofAl and A2 is given by

1A3 = AI/.2 . (22)

The thickness of the membrane is affected by the value of A3' Ifthe initial thickness of the membrane in the undeformed state is ho,then the thickness in the deformed state is

h = /.3hO' (23)

The initial thickness of a membrane was measured by placing itbetween two plastic plates. A load of 100 g was applied to the topplate. With the aid of the microscope on the micrometer slide thedistance between the two inner surfaces of the plastic plates was determined. The measurement was interpreted as the average thickness of the membrane. The average thickness of the rubber speci-mens was

h- = 1 X 10-2• (24)a

The average thickness of the fetal membranes used in the experimentswas

0.6 X 10-2 ::5 !!:. ::5 2.7 X 10-2•a

(25)



The principal strain invariants can be calculated using the expressions

11 = AI + A~ + A§

12 = AIA~ + AIA§ + A§A§

13 = AfA~A§.

ELASTIC PARAMETERS FOR THE RUBBERSPECIMEN

(26)

(27)

(28)

Traditionally rubber has been considered a Mooney material havinga strain energy function W which is represented as

(29)

Within the scope of this assumption and following the theoreticaldevelopment of Green and Adkins.t' the stress-strain relationshipsapplicable to the rubber as deflected in the experiments describedabove are

(}l = 2(Af- A§)(C 1 + A§Cz)

(}Z = 2(A~ - A§)(C1 + ArCz).

(30)

(31)

Further, Green and Adkins" show that a reasonable relation existsbetween theory and the experimental results of Treloar for rubberwhen

(32)

Using the relation of Eq. (32), one can rewrite Eqs. (30) and (31) as

iTl = 2C1(AI - A§)(l + o.n~) (33)

(}Z = 2C1(A§ - A§)(l + O.nn. (34)

Following the example of Higdon et al.,13 the longitudinal and latitudinal stresses in Eqs. (33) and (34) can be related to the kinematicsof the deflection profiles as follows:

(}1= xjaZP (dW1)-1[1+ W6(dW 1

) ZJ1/Z (35)2WoAsho dx, a Z dx,

(}Z = Ki1[~- K1UIJ. (36)A3hO

These equations result in satisfying the equilibrium conditions im-



posed on the membrane held in its deflected state by a hydrostaticpressure P.

For the "gum" rubber used in the experiments, the relation betweenthe applied pressure P and the maximum deflection Wo is shown inFig. 6. At the pole,

(37)

Also, at the pole Eqs, (33) and (34) are identical and

PITl=. (38)

2A3hoKI

Using the negative values of the first and second derivatives of Eqs.(8) and (9) derived from the function approximating the deflectionprofiles in Eq. (10),

(39)

For the values of Wo for the six modes of deflection used to generatethe points shown in Fig. 3, the longitudinal curvature K 1 and thelongitudinal stress (TI are calculated using Eqs. (38) and (39). Thevalues are placed in Eq. (34) along with a value of AI, which is equalto A2' calculated from Eq. (15) with Xi set equal to zero. A value ofC1 is obtained for each mode of deflection. The results are tabulatedin Table II.

One observes that C l varies from 0.70 kg/cm'' for the mildest modeof deflection to 1.23 kg/em? for the greatest mode of deflection. Ifthe curvature at the pole had been accurate rather than approximate,each of these values of C1 should have been the same. One considers

TABLE II

Experimental Pressures, Pole Deflections, and Membrane Thickness for theRubber Samples Along with Pole Extension Ratios and Stresses Calculated Using

Eqs. (15) and (38)a

p Wo k o Ul C1

(mm Hg) (em) (em) Al A;) (kg/cm-) (kg/cm'')

10 0.84 0.038 1.25 0.64 2.41 0.7020 1.24 0.038 1.39 0.52 4.01 1.1030 1.55 0.038 1.50 0.44 5.71 1.1340 1.80 0.038 1.61 0.39 7.39 1.2050 2.13 0.038 1.75 0.33 9.22 1.1960 2.40 0.038 1.88 0.28 11.52 1.23

a The value of C1 is calculated from Eq. (33).



that an average of the values for C1 listed in Table II should be anadequate representation of the main component of the elastic"modulus" for the "gum" rubber used in the experiments.

In this manner one assumes that the elastic parameters for the"gum" rubber used in the experiments which are independent of thevalues of II and 12 are

C1 = 1.09 kg/cm-; C2 = 0.109 kg/ern". (40)

DEFLECTION PROFILES FOR RUBBER

Although the experimental results indicate that the kinematicalresponse of the fetal membranes is the same as the kinematical response of the rubber, one is unable to determine the elastic parametersof fetal membranes using kinematical quantities obtained from theapproximated deflection function developed for rubber. However,using the approximated deflection function for rubber one can initiatean iterative procedure which in conclusion generates accurate valuesof extension ratios, first, and second derivatives, and curvatures alongthe deflection profiles. Then, if one assumes, for a given Wo, that thesekinematical quantities are the same for fetal membranes, accuratevalues for the stresses can be obtained along the deflection profile.Further, if one assumes that the fetal membranes are Mooney materials with a strain energy function represented by Eq, (29), the elasticparameters can be evaluated using Eqs. (30) and (31). A defense ofthe use of the Mooney type of strain energy function in describingbiological tissue is rendered by Blatz. 14

The iterative procedure used for the rubber membrane is now described. For the values of Wo shown in Table II, the extension ratioA2 at each value of Xi along the deflection profile is evaluated from Eq.(15). An Xj value is obtained by multiplying Xi by this value of A2.Using this value of Xj in Eq. (8), a slope is determined. The generatedslope is placed into Eq. (21) and a value of Al is obtained. The extension ratios, Al and A2' are placed into Eqs. (33) and (34) with thevalue of C, developed in Eq. (40) and A3 calculated from Eq. (22). Inthis manner, one calculates values at Xj for the longitudinal stress (O"l)and the latitudinal stress (O"z). A revised value for the slope is obtained from Eq. (35), which can be placed in the form,

(41)



In Eq. (41) the value of P and hois taken from Table II, correspondingto a given Woo This value of the first derivative is then compared tothe initial value used to generate AI. If it is not the same, the newvalue is placed into Eq. (21) and the cycle is repeated. The processis continued until the slopes generated by Eq. (41) equal the value ofthe slope applied to Eq, (22), for all the points along the deflectionprofile.

In the iteration procedure used in the research Xi was initially takenat zero and incremented in steps of 0.05until Xi equalled unity. Finalconvergence resulted after 25 cycles. However, at a point near theboundary the first term in Eq. (41)equals the value of the second termand the slope becomes infinite.

For each point where convergence of the slopes results, a latitudinalcurvature (K 2) can be calculated from Eq. (11). Using this value inEq. (36) a value for the longitudinal curvature (Kd can be generated.Rearranging terms in Eq. (36),

Kl=.l(~-(J"2K2)' (42)(Jl A3ho

In the iteration procedure it was found that as the first derivativeapproached infinity the curvature K 1 approached a constant value.Finally, a value of the second derivative can be calculated from Eq.(10) put in the form,

d2w' =K1a

2[1 + w6 (dW') 2]3/2. (43)dX j 2 Wo a 2 dx,

Using a truncated Taylor expansion at each point along the profile,starting at Xi = 0, where Wo is known, one can construct deflectionprofiles for the six experimental modes of deflection described in Fig.3 and Table II. For these six modes the deflection profiles are shownin Fig. 7. The extension ratios generating these profiles are given inFig. 8, and the longitudinal and latitudinal curvatures in Fig. 9. Finally, the longitudinal and latitudinal stresses developed in rubberalong the profiles shown in Fig. 7 are represented in Fig. 10.

In carrying out the iterative procedure the set of profiles were developed such that the final result agreed with the experimental deflection values developed from Eq. (3) using K and B values determined experimentally for rubber as shown in Table I. This wasachieved by choosing values of a, called at, in the procedure ratherthan the value of a for the membrane in the undeformed state. Therelation between the value of a 1 and a in terms of the maximum de-



.1

z<2.3

...J<{U

lll::W>;i .6zau:;~ .7::Ei5z~ .8

.9

+-BOUNDARY

Deflection Profi lesobtained from the IterativeProcedure (solid lines)compared 10ExperimentalProfiles for Rubber taorrealines ).

Experimental pointsfor Fetal Membranes givenas squares.

Experimental points forRubber given as circles.

WM' 24 mm

.21.0 ......,=---'-__-'-__..1..-_---'__-"-__--'-_--'

o

flection at the pole is given in Table III. The effectiveness of thechoice of these values is indicated by the close fit of the experimentaldata for both rubber and fetal membranes shown in Fig. 7. Further,the value of Xj for w' equal to zero is the value of a l/a. Near theboundary each theoretical curve can be completed by constructingan arc of a circle originating at a l/a and intersecting the constructedcurve at the point where the first derivative approaches infinity. Thekinematical quantities generated beyond the point where the slopesbecome infinite have no physical significance. The iterative procedure is unable to satisfy the boundary conditions at Xi = 1 since experimentally one has no knowledge of the slope and curvature of theexperimental curves at the boundary.



.4

.3

.2

1.0.4.3.2

,,,,,,,\

,,

,,

", ""''''~\\\''''...'................ ...

............. :..::~~·::~0·:~~~~lli}'Ii;%~,);r,

.9

.2

.1

.4

.6

.3

N-« .Boz~

.< .7

.5

XiFig. 8. Theoretical extension ratios for rubber (>\] and A2) as functions of the radial

distance from the pole. Al (-); A2 (- - -),

ELASTIC PARAMETERS FOR FETAL MEMBRANES

The viscoelastic nature of chorioamniotic membranes has previously been demonstrated." While being stretched in the apparatussketched in Fig. 1 the fetal membrane exhibits stress relaxation, creep,and a time-dependent stress-strain relationship. In order to obtain


MILLER, LAVERY, AND DONNELLY74

1.3

1.2

1./

10

9

.8~

3ev

l<: .7-g0

~ 63-:Z

.5

.4

.3

.2

I----~:-.:_---

,,,

,/

-------------....----................

---------------_......-

./

1.0.9.6.7.6.5

Xj

.4.3.2.1oL-_-'--_-'-_--'-_---lL-_-'--_-l..._--'-_---l'--_-'--_-'o

Fig. 9. Theoretical longitudinal (solid lines) and latitudinal (dotted lines) curvaturesfor the rubber membrane as a function of the radial distance from the pole. WM is themaximum deflection at the pole. The experimental value for WM is 24 rom. Graphsare results obtained from the iterative procedure.

experimental data between a pressure and a deflection accountableonly to the elastic nature of the membrane an experimental protocolwas devised. Each membrane was inflated by a pressure of 15 mmHg in 30 sec by extracting fluid from the lower chamber of the apparatus shown in Fig. 1. Once the necessary liquid was removed toproduce a negative pressure of 15 mm Hg the exhaust valve was closed.The pressure was then noted to relax and the relaxation was recorded



(J. -.!..(_P_) (K )-1o - 2 X3ho I xJ' 0

1.3

1.2

1.1

1.0

.9

.8

bO,ti" .7

Q

Z<l

bO .6-<0

.5

.4

.3

.2

.1

00 .2 .3 .4 .6 .7 .8 .9 1.0

Fig. 10. Computed longitudinal (solid line) and latitudinal (dotted line) stressesin the rubber membrane as a function of the radial distance from the pole. Resultsobtained by the iterative procedure.

on the physiograph. When the pressure relaxed to 10 mm Hg, thedeflection at the pole was recorded. The period of time required forthe relaxation was approximately 3 min. The exhaust valve was thenopened and liquid was returned to the lower chamber reducing thepressure to zero in a period of 30 sec. The deflection at the pole wasagain recorded. The difference in pole reading during recovery wasconsidered as the elastic pole deflection associated with a pressure



TABLE III

Values of a l Chosen for the Iterative Procedure Generating Theoretical Profiles inAgreement with Experiment"

Wo a l

(em) wolWM (em) alia

0.84 0.35 3.70 0.971.24 0.52 3.56 0.931.55 0.65 3.35 0.881.80 0.75 3.25 0.852.13 0.89 3.15 0.832.40 1.00 3.00 0.79

e WM =2.4 em, a =3.81 em.

difference of 10 mm Hg. This value of pressure was chosen in theexperiments since it is the physiologic resting pressure in the uterusduring pregnancy. The measurement of the membrane thickness wascarried out as described previously. The pressure difference duringrecovery (P), the recoverable extension at the pole (wo),and the initialthickness of the membrane (ho) are used to determine the elasticparameters of the membrane. Using the measured value of Wo acorresponding pressure for rubber is obtained from the relation shownin Fig. 6. Using this pressure, the pole deflection, and the thicknessof the rubber sample (0.038 em), one calculates the extension ratios,first and second derivatives, and curvatures using the iterative procedure previously described. In this procedure the aforementionedelastic parameters for rubber are used in these calculations. The iteration is carried out for Xi from zero to 0.5 in increments of 0.1. Theiterated derivatives, and curvatures are placed in Eqs. (35) and (36)along with the pressure of 10 mm Hg obtained in the experiment withthe fetal membranes and the experimental Wo and h o for the fetalmembranes. The longitudinal and latitudinal stresses calculated arethen placed into Eqs, (30) and (31) along with the extension ratioscalculated by the iterative procedure using pressures related to a Woaccording to Fig. 6. The twoequations are then solvedsimultaneouslyfor C1 and C2. These values represent the elastic parameters for thefetal membranes associated with a pressure differential of 10 mmHg.

Experiments and calculations were carried out for 55 specimensof fetal membranes obtained after birth from pregnancies which cameto term. The value of C1 and C2 calculated for each membrane wasthe same at each value of Xi from zero to 0.5. Also each value of Cz



was 0.1, the value of C1. The distribution of the value of C1 to thenearest 0.1 kg/cm'' is shown in Fig. 11 for the 55 membranes. Thelargest value of C1 recorded was 1.5 kg/em? and the smallest value was0.1 kg/cm'', The population peaked at C1 = 0.5 kg/em",

DISCUSSION

The theoretical equations describing the deflection profiles andthe extension ratios produce kinematical values which agree equallywith the rubber and the fetal membrane experiments. The experimental deflection profiles are quite different from the profiles generated by Treloar'' which are rather "spherical" in nature. However,the differences are understandable when one considers that Treloarexpanded his membranes into an unbounded space; whereas in theresearch presented here, the membranes are expanded within theconfines of a cylindrical tube. Such an expansion is more physiological to fetal membranes expanding within the confines of the cervical cavity. Also, one can expect the extension ratios and the curvatures to be different from the previous theoretical studies describedin Green and Adkins' text.s

The fact that the theoretical profiles developed in the research donot satisfy the boundary conditions does not vitiate the ultimate goalof the research-the establishment of a method to determine elasticparameters for fetal membranes. In the region from Xi = 0 to Xi =

20

1a 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

(GI) IN Kg/CM2

Fig. 11. The distribution of the principal elastic parameter (Ctl for the fetalmembranes taken from term pregnancies.



0.5 the extension ratios and stresses have their maximum values andgreatest variation. However, it is within this region that the iteratedvalues of the profiles agree with the experimental values. Also, thederivatives and curvatures generated using rubber are applicable todetermining stresses in fetal membranes. Hence, in the developmentof elastic parameters for fetal membranes, one uses the kinematicalresults for rubber in the same manner as one uses the results fromstrain gauges in analyzing stresses in metals.

It seems reasonable to conclude that the values obtained for C1

using 55 fetal membrane specimens adequately represent the principalelastic parameters for the term membranes. Therefore, it appearspromising that using this method of calculation, one will be able tomake a reasonable comparison of the elastic parameters generatedfor membranes from term pregnancies to those membranes whichrupture at various gestational stages.

The authors gratefully acknowledge the financial support of the United CerebralPalsy Research Foundation and the William Randolf Hearst Foundation in the conductof this research. Also, they wish to thank their students, Audrey Spencer, MarcellaDenton, Debbie Smither, and Hans Fiedler, for their services in the conduct of theexperiments and calculations.

References

1. G. C. Gunn, D. R. Mishell, and D. G. Morton, Am. J. Obstet. Gynecol., 196,469(1970).

2. R. Artal, Am. J. Obstet. Gynecol., 125,655 (1976).:J. D. N. Danforth, Am. J. Dbstet. Gynecol., 65,480 (1953).4. T. B. MacLachlan, Am. J. Obstet. Gynecol., 91,309 (1965).5. W. Z. Polishuk, J. Obs. Gyn. Brit. Comm., 83,422 (1976).6. E. Parry-Jones, Brit. J. Obstet. Gynecol., 83,205 (1976).7.•J. P. Lavery and C. E. Miller, Obstet. Gynecol .. 50,467 (1977).8. A. E. Green and J. E. Adkins, Large Elastic Deformations, Clarendon Press,

Oxford, England, 1970, Chap. 4.9. R. S. Rivlin, Phil. Trans. Roy. Soc. A, 379 (1948).

10. T. A. Donnelly, M. Eng. thesis, Univ. of Louisville, 1977.11. A. L. Schwartz, Am. J. Obstet. Gynecol. 127,470 (1977).12. W. H. Yang and W. W. Feng, J. Appl. Mech., 1003-1011 (1970).13. A. Higdon, E. H. Olsen, W. B. Stiles et aI.,Mechanics of Materials, John Wiley,

New York, 1976, p. 152.14. P. J. Blatz, Trans. Soc. Rheol., 13,83 (1969).

Received November 2,1977Accepted as revised June 22,1978


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Determination of Elastic Parameters For Human Fetal Membranes