1. I. Background and Significance
Parturition, known colloquially as childbirth, is the process by which a developed fetus is
delivered through the cervix at the end of pregnancy. This process is stimulated largely through
the actions of a hormone called oxytocin, which interacts with oxytocin receptors in the uterus to
stimulate uterine contractions.2 Oxytocin has been nicknamed the “love hormone” due to the role
it plays not only in parturition, but also in mammary gland activity during breast suckling and
various other maternal behaviors, as well as sexual orgasm.3 In order to prevent damage to the
fetus and mother from unnecessary contractions the effects of oxytocin on the uterus are
normally inhibited by neuroactive steroid metabolites of progesterone, and specifically inhibited
by a central opioid inhibitory mechanism as pregnancy is coming to term.4-6 As pregnancy is
near term the secretion of progesterone is significantly decreased, and the central opioid
mechanism can be overcome through the oxytocin concentration within the uterus reaching a
threshold. 5,6
In addition to the lessening of endogenous inhibitory systems to oxytocin, the
concentration of oxytocin receptors within the uterus also increases dramatically in the final
stages of pregnancy.7,8 This sharply increases the uterus’ sensitivity to oxytocin in preparation
for parturition. Mechanical stretch receptors that signal for the release of oxytocin also appear in
greater concentrations in the cervical wall during this period.9,10 A schematic of the fetus in utero
is shown in Figure 1.
2. Parturition is finally initiated by uterine pressure forcing the fetus’ head against the
cervical wall.2,11 The cervical wall tissue houses mechanical stretch receptors, which signal for
the release of oxytocin in proportion to the magnitude of the achieved stretch. This stretching
occurs in the form of cervical dilation, in which the cervix is eventually opened allowing for
final delivery of the fetus.11 The neural impulse from the receptors on the cervical wall is
delivered to the hypothalamus, and stored oxytocin is released into the blood from the posterior
pituitary. 2,8
As plasma oxytocin concentration increases, assuming the initial burst of oxytocin is
great enough to overcome the central opioid inhibitory mechanism, oxytocin binds to oxytocin
receptors in the uterus, stimulating a contraction of the uterine wall. The amplitude of uterine
contraction is dependent on the concentration of oxytocin in the uterine plasma.12,13 The
oxytocin-induced contraction pushes the fetus further into the cervical wall, causing additional
dilation of the cervix. This causes the cervix’s stretch receptors to signal for the release of more
oxytocin, stimulating further contractions, which induces additional cervical stretching, and the
cycle continues. This positive feedback loop continues until the fetus is expelled out of the cervix
through the vagina, after which the stimulus for the release of oxytocin disappears.2
Figure 1: Anatomy of
Fetus in Utero1
3. Common complications during parturition include the inability to overcome the central
opioid inhibition mechanism and outlying trauma to the fetus or mother. In the previous case,
injection of exogenous oxytocin is often prescribed in order to “kick-start” parturition, as is the
application of a cesarean section.14,15 In the latter case, parturition can either be delayed by
exogenous injection of opioids, thus reasserting the influence of the central inhibition
mechanism,6,16 or completed through a cesarean section. A quantitative model capable of
predicting the need for these procedures would reduce the risk of injury to or loss of the fetus
during parturition. The model we have built modeling parturition in the ideal case, where no
trauma occurs to the mother or fetus and the central opioid inhibition mechanism is overcome,
could be used as a starting point for such a model.
IIa. System Properties
The system properties, including their designation, physiological relevance and
quantitative values, are shown in Table 1 below. The quantitative values for these properties
were derived from empirical data presented in the literature sources cited.
Designation Physiological Significance Quantitative Value
k1 (Initial diameter of cervix) /
Elastic Modulus of Cervical
Wall over time, rate of
Cervical dilation vs. Pressure
12 * 10-6 cm/(mmHg * s) 17,18
k2 Elastic recoil of Cervical Wall N/A (ignored term)
k3 Ratio of Pressure of Oxytocin-
induced contractions vs.
concentration of Oxytocin in
pg/mL
57.9 *10-6
(pg/mL)/(cm*s)12,17-19
k4 Ratio of Possible Additive
Pressure from “pushing” vs.
Pressure of contractions
induced by Oxytocin
0.487 mmHg/mmHg20,21
k5 Elastic recoil of Uterine Wall N/A (ignored term)
4. k6 Viscosity of Amniotic Fluid N/A (ignored term)
k7 Ratio of rate of release of
oxytocin vs. mechanical
stretch (dilation) of uterus
19.3 (pg/mL)/(cm*s)12,19
k8 Term describing rate of
endogenous clearance of
Oxytocin
0.001155 s-1 22
Table 1: System properties of our proposed quantitative model of parturition.
IIb. System Variables
We analyzed three variables in the process of generating our quantitative model.
1: Cervical pressure, or fetus head-to-cervix pressure, was measured in mmHg. This pressure is
generated by the intrauterine pressure, which is itself generated by a combination of oxytocin-
induced contractions and the mother’s additional “pushing.” Cervical pressure has been found to
generally be roughly four times the intrauterine pressure;11 this was taken into account when
deriving our value for k1. The viscosity of the amniotic fluid and the recoil tendency of the
uterine wall counteract this generation of pressure, however for mathematical convenience these
factors were ignored.
2: Mechanical Stretch of the Cervical Wall, or dilation of the cervical wall, was measured in
centimeters. The mechanical stretch of the cervical wall is generated directly by the cervical
pressure applied to the cervix. It has been shown that the magnitude of the neural signaling from
the stretch receptors on the cervical wall is directly related to the mechanical stretch of the wall;
2,17this has been taken into account in the form of the system property k7. The cervical wall, like
any elastic element, has a tendency to recoil when stretched, counteracting the generated dilation
of the cervical wall. However, this was ignored during the generation of our model for
mathematical convenience.
5. 3: Plasma Concentration of Oxytocin in the Uterus was measured in pg/mL. The plasma
concentration of oxytocin in the uterus determines the number of activated oxytocin receptors
within the uterus, and as such the concentration of oxytocin is directly related to the amplitude of
uterine contractions. This plasma concentration is generated by the release of oxytocin from the
posterior pituitary into the blood, as stimulated by the stretch receptors on the cervical wall
during parturition.2 Oxytocin is also endogenously cleared from the bloodstream over time,
partly due to binding with uterine oxytocin receptors, partly due to blood flow and partly due to
the central opioid inhibition mechanism.22
As explained by our Block Diagram in an upcoming section, we took our system input to
be the Cervical Dilation and our system output to be Cervical Pressure.
IIc. SystemAssumptions
The assumptions we made in generating our quantitative model are as follows:
1: The central opioid inhibition mechanism is successfully overcome with the initial release of
oxytocin.
2: Positive Feedback, and hence the application of our model, begins after the first Oxytocin-
induced contraction, which we considered as the initiation of parturition.
3: The recoil tendency of the cervical wall was ignored for mathematical convenience.
4: The recoil tendency of the uterine wall was ignored for mathematical convenience.
5: The effect of the viscosity of the amniotic fluid on the intrauterine pressure was ignored for
mathematical convenience.
6: The effective area for the relationship between the cervical wall and the fetus head was
considered to be time-invariant. Assuming this made is simpler to use the elastic modulus of the
cervical wall to calculate the mechanical stretch/dilation of the wall over time.
7: The cervical pressure is assumed to constantly be exactly four times the intrauterine pressure.
8: The stress/strain behavior of the cervical wall is assumed to be defined solely by its elastic
modulus.
9: The sensitivity of the uterine oxytocin receptors to oxytocin is assumed to be time-invariant.
6. III. Block Diagram
The block diagram we generated to begin to analyze the parturition system is shown below.
*Note: The Mechanical Stretch/Dilation of the Cervical Wall is directly related to the neural
stimulus for the release of oxytocin. While the sensor action and input is physiologically this
neural stimulus, we decided to use the mechanical stretch/dilation of the cervix for this purpose
for mathematical convenience.
IV. Mathematical Analysis of System Response to Unit and Periodic Input Functions
The following section outlines the procedure we followed while developing our system, finding
its transfer function, and determining unit impulse and frequency response.
𝑑𝐷(𝑡)
𝑑𝑡
= 𝑘1 𝑃( 𝑡) − 𝑘2 𝐷(𝑡) (1)
𝑑𝑃(𝑡)
𝑑𝑡
= 𝑘3 𝐶( 𝑡) + 𝑘4 𝑃( 𝑡) − 𝑘5 𝑃( 𝑡) − 𝑘6 𝑃(𝑡) (2)
𝑑𝐶(𝑡)
𝑑𝑡
= 𝑘7 𝐷( 𝑡) − 𝑘8 𝐶(𝑡) (3)
Taking the LT of (1),(2) and (3)
𝑠𝐷( 𝑠) − 𝐷(0) = 𝑘1 𝑃(𝑠) (4)
𝑠𝑃( 𝑠) − 𝑃(0) = 𝑘3 𝐶( 𝑠) + 𝑘4 𝑃(𝑠) (5)
𝑠𝐶( 𝑠) − 𝐶(0) = 𝑘7 𝐷( 𝑠) − 𝑘8 𝐶(𝑠) (6)
Assuminginitial conditionsas 0, we get
𝑠𝐷( 𝑠) = 𝑘1 𝑃(𝑠) (7)
𝑠𝑃( 𝑠) = 𝑘3 𝐶( 𝑠) + 𝑘4 𝑃(𝑠) (8)
+
Uterine
Contraction/Cervical
Wall Pressure
(Effector)
Hypothalamus/
Posterior Pituitary
(Plant)
Uterus
(Sensor)
Cervical WallMechanical Stretch/Dilation
of Cervical Wall*
Release of Oxytocin
Central Opioid
Inhibition Mechanism
(Disturbance)
Labor Initiation from contact
between fetus and cervical wall
X
8. 𝑥0 𝑒𝑗𝑤𝑡
=
𝑧𝑗2 𝑤2 𝑒 𝑗𝑤𝑡
𝑘3 𝑘7
−
𝑘4 𝑧𝑗𝑤𝑒 𝑗𝑤𝑡
𝑘3 𝑘7
+
𝑘8 𝑧𝑗𝑤𝑒 𝑗𝑤𝑡
𝑘3 𝑘7
−
𝑘8 𝑘4 𝑧𝑒 𝑗𝑤𝑡
𝑘3 𝑘7
(24)
Diving (24) by z,
𝑥0
𝑧
=
𝑗2 𝑤2
𝑘3 𝑘7
−
𝑘4 𝑗𝑤
𝑘3 𝑘7
+
𝑘8 𝑗𝑤
𝑘3 𝑘7
−
𝑘8 𝑘4
𝑘3 𝑘7
(25)
Simplifying(25),
𝑥0
𝑧
=
𝑗2 𝑤2−𝑘4 𝑗𝑤+𝑘8 𝑗𝑤−𝑘8 𝑘4
𝑘3 𝑘7
or,
𝑧
𝑥0
=
𝑘3 𝑘7
𝑗2 𝑤2−𝑘4 𝑗𝑤+𝑘8 𝑗𝑤−𝑘8 𝑘4
(26)
If
𝒛
𝒙 𝟎
= 𝑯 𝟎( 𝒘) and 𝑯 𝟎(𝒘) = |𝑯 𝟎( 𝒘)|𝒆𝒋(𝒘𝒕+ɸ( 𝒘)), pluggingin,
|𝐻0( 𝑤)| =
𝑘3 𝑘7
√(−𝑤2−𝑘4 𝑘8)2+((−𝑘4+𝑘8)𝑤)2
(27)
And
ɸ( 𝑤) = −tan−1
(
(−𝑘4+𝑘8)𝑤
−𝑤2−𝑘4 𝑘8
) (28)
AssumingX0=1, we get
𝑧 =
𝑘3 𝑘7
√(−𝑤2−𝑘4 𝑘8)2+((−𝑘4+𝑘8)𝑤)2
𝑒
𝑗(𝑤𝑡−tan−1(
(−𝑘4+𝑘8) 𝑤
−𝑤2−𝑘4 𝑘8
))
(29)
System Response to Unit Impulse
Using (14), we can find our system’s response to unit impulse.
𝑃( 𝑡) =
𝑘3 𝑘7 𝑒 𝑘3 𝑡
𝑘3+𝑘4
−
𝑘3 𝑘7 𝑒−𝑘8 𝑡
𝑘3+𝑘4
(14)
We plotted this function over time using a simple plotting algorithm in Matlab. The resulting plot
is shown on the next page in Figure 2. The general form of this equation, a summation of an
exponential increase and an exponential decay, is justifiable physiologically considering the
interactions between the released oxytocin from the posterior pituitary and the central opioid
inhibitory mechanism.
9. As expected, the system responds to a unit impulse with an exponential increase over
time, indicative of the positive feedback present within the system. The maximum value for
cervical pressure after 1000 minutes, or roughly 16 and two-thirds hours, is equal to 352 mmHg.
This correlates well with the maximum observed value for cervical pressure in literature during
the same time period, roughly 320 mmHg.
System Response to Periodic Input
Using (29), and assuming the amplitude of our input function Xo =1, the frequency
response of our system is:
𝑧 =
𝑘3 𝑘7
√(−𝑤2−𝑘4 𝑘8)2+((−𝑘4+𝑘8)𝑤)2
𝑒
𝑗(𝑤𝑡−tan−1(
(−𝑘4+𝑘8) 𝑤
−𝑤2−𝑘4 𝑘8
))
(29)
The Frequency Response of our system operating under several angular frequencies is shown in
Figure 3a-d.
Figure 2: System Unit Impulse Response
10. Figure 3a: System Frequency Response, ω = π.
Figure 3b: System Frequency Response, ω = 1.5π.
11. As shown by Figure 3a-d, our system responding to a sinusoidal with a sinusoidal output. As
expected from the nature of our coefficients, the amplitude of the response is much smaller than
the amplitude of the input. It appears that with increasing angular frequency ω our system
responds with decreased amplitude and a decreased phase lag.
Figure 3c: System Frequency Response, ω = 2π.
Figure 3d: System Frequency Response, ω = 4π.
12. V. Stability Analysis
Analyzing the stability of the transfer function, from (13)
𝑃(𝑠)
𝐷(𝑠)
=
𝑘3 𝑘7
(𝑠+𝑘8)(𝑠−𝑘4)
=
( 𝑘3 𝑘7)
𝑠2+( 𝑘3−𝑘4) 𝑠−𝑘3 𝑘4
(13)
the system presented is unstable according to the Routh-Hurwitz necessary stability criteria.
While this system fulfills the first necessary condition, as none of the coefficients of the relevant
polynomial in the denominator are equal to zero, if does not satisfy the condition stating that all
coefficients of the polynomial must be of the same sign. Also, it is plainly obvious that the
relevant polynomial of this system possesses a root with a real positive part, another sign of
instability.
VI. Conclusions
It is generally accepted that the release of oxytocin during parturition is performed
physiologically as a positive feedback loop, and when modeled using the parameters we have
chosen the responses to unit impulse and frequency response showed properties of positive
feedback behavior. The behavior of the plot of cervical pressure over time due to unit impulse
shows an exponential increase. This is expected of a system featuring positive feedback, as the
feedback of the system constantly increases the output with every cycle. When the system input
changes in a sinusoidal fashion, the system responds in a similar sinusoidal fashion, however
with a decrease in both amplitude phase lag as the input frequency increases.
However, our model is only a partial model of parturition. It stays within the boundaries
of our assumptions and only models an ideal case where the system both overcame the central
opioid inhibition mechanism and no trauma to the fetus or mother was introduced to the system.
In a physiological situation, properties such as the maximum cervix dilation and maximum
uterine contractile response to oxytocin will influence the results and behavior of the entire
13. system. Had the model incorporated such limits, the cervical pressure would reach a maximum
as determined by the maximum dilation and maximum uterine contractile response to oxytocin.
A predictive mathematical model such as the one we have designed can be used for future
studies to reduce the risk of injury to either the fetus or the mother. Limits such as the effective
area of the fetal head, maximum dilation and pressure response can be added to the quantitative
model to bind the responses. Finally, the system properties and parameters we ignored in our
analysis can also be added to simulate a more realistic case for future analysis physiological
control system of human parturition.
VII. References
1. Pittsburg CsHo. Anatomy: Fetus in Utero; 2013.
2. Russell J, Leng G, Douglas A. The magnocellular oxytocin system, the fount of
maternity: adaptations in pregnancy. Frontiers in Neuroendocrinology. 2003;24(1):27–61.
3. Lee H, Macbeth, Young W. Oxytocin: The great facilitator of life. Progress in
Neurobiology. 2009;88(2):127–151.
4. Antonijevic I, Douglas A, Dye S, Bicknell R, Leng G, Russell J. Oxytocin antagonists
delay the initiation of parturition and prolong its active phase in rats. J Endocrinol. 1995;145:97-
103.
5. Antonijevic I, Russell J, Bicknell R, Leng G, Douglas A. Effect of progesterone on the
activation of neurones of the supraoptic nucleus during parturition. J Reprod Fertil.
2000;120:367-376.
6. Douglas A, Clarke G, Macmillan S, et al. Effects of the k-opioid against U50,488 on
parturition in rats. Br J Pharm. 1993;109:251-258.
7. Russell J, Leng G. Sex, parturition and motherhood without oxytocin? J Endocrinol.
1998;157:343-359.
8. Higuchi T, Tadokoro Y, Honda K, Negoro H. Detailed analysis of blood oxytocin levels
during suckling and parturition in the rat. J Endocrinol. 1986;110:251-256.
9. Terzidou V, Sooranna SR, Kim LU, Thornton S, Bennett PR, Johnson MR. Mechanical
stretch up-regulates the human oxytocin receptor in primary human uterine myocytes. J Clin
Endocrinol Metab. 2005;90(1):237-246.
10. Loudon J, Sooranna S, Bennett P, Johnson M. Mechanical stretch of human uterine
smooth muscle cells increases IL-8 mRNA expression and peptide synthesis. Molecular Human
Reproduction. 2004;10(12):895-899.
11. Antonucci MC, Pitman MC, Eid T, Steer PJ, Genevier ES. Simultaneous monitoring of
head-to-cervix forces, intrauterine pressure and cervical dilatation during labour. Med Eng Phys.
1997;19(4):317-326.
12. Yusoff D, Raghavan L, Pociask C, Fuchs, F. Oxytocin in Human Pregnancy and
Parturition : Obstetrics & Gynecology. Obstetrics and Gynecology. 1978;51(2).
14. 13. Kawamata M, Tonomura Y, Kimura T, Yanagisawa T. The differential coupling of
oxytocin receptors to uterine contractions in murine estrous cycle. Biochemical and Biophysical
Research Communications. 2004;321(3):695–699.
14. Al-Eknah M, Noakes D. The effect of different rates of oxytocin infusion on the
progesterone primed uterus in chronically catheterised, ovariectomised ewes. Animal
Reproduction Science. 1988;17(Issues 1–2):69–76.
15. Miller F, Mattison D. Oxytocin for induction of labor: a critical evaluation. Endocrine
Disorders in Pregnancy. 1989:411-425.
16. Goodwin T, Paul R, Silver H, et al. The effect of the oxytocin antagonist atosiban on
preterm uterine activity in the human * ** * **. American Journal of Obstetrics and
Gynecology. 1994;170(2):474–478.
17. Dewall RJ, Varghese T, Kliewer MA, Harter JM, Hartenbach EM. Compression-
Dependent Viscoelastic Behavior of Human Cervix Tissue. Ultrason Imaging. 2010;32(4):214-
228.
18. Hernandez-Andrade E, Hassan SS, Ahn H, et al. Evaluation of cervical stiffness during
pregnancy using semiquantitative ultrasound elastography. Ultrasound Obstet Gynecol.
2013;41(2):152-161.
19. O'Byrne KT, Ring JP, Summerlee AJ. Plasma oxytocin and oxytocin neurone activity
during delivery in rabbits. J Physiol. 1986;370:501-513.
20. Buhimschi CS, Department of Obstetrics GaRS, The University of Maryland School of
Medicine, Baltimore, MD, USA, Buhimschi IA, et al. The effect of fundal pressure manoeuvre
on intrauterine pressure in the second stage of labour. BJOG: An International Journal of
Obstetrics & Gynaecology;109(5):520-526.
21. Buhimschi C, Buhimschi I, Malinow A, Weiner C. Intrauterine Pressure During the
Second Stage of Labor in Obese Women. Obstetrics and Gynecology. 2004;103(2):225-230.
22. Lobo M, Bastos L, Meurs W, Campos D. A model for educational simulation of the
effect of oxytocin on uterine contractions. Medical Engineering & Physics. 2013;35(4):524–531.
