Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
A biomechanical model for the gravid uterus
B. Irfanoglu & E. Karaesmen
Department of Engineering Sciences, Middle East
Technical University, 06531 Ankara, Turkey
ABSTRACT
The biomechanics of the uterus is studied on a mathematical model based
on a shell of ellipsoid of revolution. The myometrium is treated as a
homogenous, isotropic and incompressible elastic material using
continuum approach and large deformation theory as introduced by Fliigge
and Chou [1] and Green and Adkins [2]. The governing equations which
are nonlinear differential equations of boundary value type are solved by
the shooting method together with the Runge-Kutta integration scheme.
The numerical results showing the variation of stretch ratios with uterine
pressure are presented graphically for neo-Hookean and Mooney-Rivlin
type of materials.
INTRODUCTION
The importance of mechanical parameters in uterine activity has been well
recognized but, unfortunately, manifested only in relatively small number
of related publications. Mirrahi, Kami and Polishuk [3] have presented a
brief survey of relevant literature and proposed a thin shell model for the
uterus. In their paper, the solution of the deformed shell is based on the
incremental states of deformation where each deformation is sufficiently
small in itself. They used in vivo multi-point strain measurements as
boundary conditions for their numerical solutions and presented changes
in curvature and local obliqueness of the uterus during contraction.
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
60
Computational Biomedicine
In the present study, the uterus deformation is considered as a large
deformation problem and the theoretical model is developed both for
geometrical and material nonlinearities using thin shell theory. Initially
spheric membrane type of modelling under these assumptions has been
used for the hernical sac deformation by Engin and Akka§ [4]. Here,
the pear like shape of the uterus [5] is approximated by an ellipsoid of
revolution. During pregnancy there occurs great growth of the walls of
uterus due to estrogens, so there must be a biochemical foundation for the
molecule by molecule change and grow of the cells and tissues. But it must
have a biophysical foundation, also [6, 7]. This paper is concerned with the
effects of stress and strain and aims to present a mechanical model that
can serve as a means to help discriminate between the possible suggested
mechanisms and to evaluate the driving force on the fetal head which
becomes important in the cases of lower uterine spasm. The influence of
the material properties of the uterus, its thickness, its initial curvature
and the amniotic-fluid pressure are obtained numerically and presented
graphically.
THEORETICAL MODEL AND GOVERNING EQUATIONS
The myometriurn will be modeled using continuum approach like a
membrane experiencing large deformations but will have no rigidity
against bending. At all stages of pregnancy, which is the scope of the
present study, the membrane is assumed to be clamped around the
undeformable and stiff cervix. In related previous research work it was
pointed out [3] that isotropy prevails even in most of the stages of labor,
except the final stage in the uterine muscle. The mathematical model in
the form of an ellipsoid of revolution as shown in Figure 1, is thus assumed
to be made of isotropic, homogeneous and incompressible material.
The equilibrium equations for the large deformations and rotations
of a shell of revolution are available in Flugge and Chou [1]. The
nondimensional form of the governing equations under the aforementioned
assumptions in terms of undeformed reference configuration (f, r\, </>) are
as follows
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
TV*
e i
- — -- simp
AT^r
61
(la)
^ /
r
-1
o
i
-7— = — \<hcosy -- \ecos(p
d(p
r
r
(Ic)
For a Mooney-Rivlin type of material whose strain energy density
function W is defined in terms of the strain invariants /i and /2 as
W = Ci(/i-3) + C2(/2-3)
(2)
the forces per unit length of shell of revolution N$ and N$ in the meridional
and circumferential directions, respectively, become
(3a)
(36)
The nondimensional quantities in the above set of equations are
obtained through the following equalities
(4)
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
62
Computational Biomedicine
where P is the internal amniotic fluid pressure, r,ri,0 and 0 are the
geometrical quantities which describe the undeformed configuration of
the shell as shown in Figure 1, A^ and A# are the stretch ratios in the
meridional and circumferential directions, respectively. In the case of an
ellipsoid of revolution, in terms of the major and minor radii a and b, r%
becomes
(5)
When the constitutive Eqs. (3) and the geometrical consideration
given in Eq. (5) are substituted in Eqs.(l), the elimination of Ne and A/^,
will result in three first-order highly nonlinear differential equations for
the unknowns A# and \ and ^.
Deformed
Configuration
Reference
Configuration
Figure 1. Geometric model
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
63
NUMERICAL ANALYSIS AND RESULTS
The governing equations, which are subject to the constraint
condition that the radius of the lower opening of the deformed uterus
must always be equal to the radius of the cervix are thus of boundary
value type of differential equations. The two other conditions which are
necessary for the solution of the equations are obtained from the axial
symmetry properties of the problem, namely at (j) = 0°, it is known that
i/> — 0° and A# = \<$> = A<>. Using the shooting method approach for
the numerical analysis and for the independent variables <f> and p, the
progresssing deformation of the uterus in terms of the stretch ratios A#
and A<£ are obtained. In Figure 2 for a given cervix opening and initial
configuration the change of the apex stretch ratio A<>, with respect to the
amniotic pressure is shown for Mooney-Rivlin material and for (a =.0.1)
Neo-Hookean type of material (a = 0.0). The change of total volume
enclosed with respect to the pressure for the same geometry and material
properties are shown in Figure 3. The initially pressurized configuration is
ellipsoid of revolution, but the stretch ratio is assumed to be very slightly
larger than unity and during deformations the membrane is further
assumed to be free of wrinkling ( no compressive stresses ).
One of the questions that may be answered by the aid of the present
model is the magnitude of strains and stresses in the uterus walls at various
states of pregnancy. Since the nonlinear elastic properties of the walls
are described by the strain energy density function which has almost an
infinite nuber of possible forms [8, 9] we are capable to reproduce any
experimental data.
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
64
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Figure 2. Change of nondimensional pressure Pn with respect to the
stretch ratio AO at <j> = 0° (a = 0.0 : Neo-Hookean material,
a = 0.1 : Mooney-Rivlin type of material; r\ = 5.0, £ = 1.5).
3.5
3.0 -
a - 0.0
2.5 2.0 1.5 1.0 0.5 0.0
0.0
0.5
1.0
1.5
2.0
2.5
Pn
Figure 3. Pressure versus volume enclosed during deformation for the
same parameters given in Figure 2.
Transactions on Biomedicine and Health vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-3525
Computational Biomedicine
65
REFERENCES
1. Fliigge, W. and Chou, S.C. 'Large Deformation Theory of Shells of
Revolution', J. Applied Mechanics, Vol.34, pp. 56-58, 1967.
2. Greenm A.E. and Adkins, J.E. Large Elastic Deformation
Nonlinear Continuum Mechanics, Oxford University Press, 1960.
and
3. Mizrahi, J., Kami, Z. and Polishuk, W. 'A Kinematic Analysis of
Uterine Deformation During Labor', J. The Franklin Inst. Vol.306, No.2,
pp. 119-132, 1978.
4. Engin, A.E, and Akka§, N. 'Etiology and Biomechanics of Hernial Sac
Formation', J. Biomed. Eng., Vol. 5, pp-329-335,1983.
5. Netter, F. Reproductive System, Vol.11, Ciba Collection of Medical
Illustrations, New York, 1965.
6. Fung Y.C. Biomechanics : Motion, Flow, Stress and Growth, SpringerVerlag, New York, 1990.
7. Zinemanas, D. and Nir, A. 'A Fluid-Mechanical Model of Deformation
During Embryo Exogastrulation', J. Biomechanics, Vol.25, No.4, pp.341346,1992.
8. Fung, Y.C 'Biorheology of Soft Tissues', J.Biorheology Vol.10, pp. 139155, 1973.
9. Demiray, H. 'Finite Elasticity of Soft Biological Tissues', J. Pure and
Applied Sciences, Special Bio-Engineering Issue, pp. 44-51, 1977.