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.

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

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)

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.

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. System Assumptions

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.

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= 𝑘1𝑃(𝑡)− 𝑘2𝐷(𝑡) (1)

𝑑𝑃(𝑡)

𝑑𝑡= 𝑘3𝐶(𝑡)+ 𝑘4𝑃(𝑡) − 𝑘5𝑃(𝑡)− 𝑘6𝑃(𝑡) (2)

𝑑𝐶(𝑡)

𝑑𝑡= 𝑘7𝐷(𝑡) − 𝑘8𝐶(𝑡) (3)

Taking the LT of (1), (2) and (3)

𝑠𝐷(𝑠) − 𝐷(0) = 𝑘1𝑃(𝑠) (4)

𝑠𝑃(𝑠)− 𝑃(0) = 𝑘3𝐶(𝑠) + 𝑘4𝑃(𝑠) (5)

𝑠𝐶(𝑠)− 𝐶(0) = 𝑘7𝐷(𝑠)− 𝑘8𝐶(𝑠) (6)

Assuming initial conditions as 0, we get

𝑠𝐷(𝑠) = 𝑘1𝑃(𝑠) (7)

𝑠𝑃(𝑠) = 𝑘3𝐶(𝑠) + 𝑘4𝑃(𝑠) (8)

+ Uterine

Contraction/Cervical

Wall Pressure

(Effector)

Hypothalamus/ Posterior Pituitary

(Plant)

Uterus

(Sensor)

Cervical Wall Mechanical Stretch/Dilation

of Cervical Wall*

Release of Oxytocin

Central Opioid

Inhibition Mechanism

(Disturbance)

Labor Initiation from contact between fetus and cervical wall

X

𝑠𝐶(𝑠) = 𝑘7𝐷(𝑠)− 𝑘8𝐶(𝑠) (9)

Solving for C(s) and P(s),

𝐶(𝑠) =𝑘7𝐷(𝑠)

𝑠+𝑘8 (10)

𝑃(𝑠) =𝑘3𝐶(𝑠)

(𝑠−𝑘4) (11)

Plugging (10) into (11),

𝑃(𝑠) =𝑘3𝑘7𝐷(𝑠)

(𝑠+𝑘8)(𝑠−𝑘4) (12)

Solving for 𝑷(𝒔)

𝑫(𝒔) in (12),

𝑃(𝑠)

𝐷(𝑠)=

𝑘3𝑘7

(𝑠+𝑘8)(𝑠−𝑘4) (13)

Taking the Laplace Inverse after setting D(s) = 1, we obtain a system response to unit impulse,

𝑃(𝑡) =𝑘3𝑘7𝑒

𝑘3𝑡

𝑘3+𝑘4−

𝑘3𝑘7𝑒−𝑘8𝑡

𝑘3+𝑘4 (14)

--------------

Differentiating (2), and solving for 𝒅𝑪(𝒕)

𝒅𝒕,

𝑑2𝑃(𝑡)

𝑑𝑡2= 𝑘3

𝑑𝐶(𝑡)

𝑑𝑡+ 𝑘4

𝑑𝑃(𝑡)

𝑑𝑡 (15)

𝑑𝐶(𝑡)

𝑑𝑡=

1

𝑘3

𝑑2𝑃(𝑡)

𝑑𝑡2−

𝑘4

𝑘3

𝑑𝑃(𝑡)

𝑑𝑡 (16)

Substituting (16) back into (3),

1

𝑘3

𝑑2𝑃(𝑡)

𝑑𝑡2−

𝑘4

𝑘3

𝑑𝑃(𝑡)

𝑑𝑡= 𝑘7𝐷(𝑡)− 𝑘8𝐶(𝑡) (17)

Separating (2) and solving for C(t),

𝑑𝑃(𝑡)

𝑑𝑡= 𝑘3𝐶(𝑡)+ 𝑘4𝑃(𝑡) (18)

𝐶(𝑡) =1

𝑘3

𝑑𝑃(𝑡)

𝑑𝑡−

𝑘4

𝑘3𝑃(𝑡) (19)

Substituting (19) into (17) and solving for D(t),

𝐷(𝑡) =1

𝑘3𝑘7

𝑑2𝑃(𝑡)

𝑑𝑡2−

𝑘4

𝑘3𝑘7

𝑑𝑃(𝑡)

𝑑𝑡+

𝑘8

𝑘3𝑘7

𝑑𝑃(𝑡)

𝑑𝑡−

𝑘8𝑘4

𝑘3𝑘7𝑃(𝑡) (20)

𝑃(𝑡) = 𝑧𝑒𝑗𝑤𝑡 (21)

𝐷(𝑡) = 𝑥0𝑒𝑗𝑤𝑡 (22)

Substituting (21) and (22) into (20),

𝑥0𝑒𝑗𝑤𝑡 =

1

𝑘3𝑘7

𝑑2(𝑧𝑒𝑗𝑤𝑡)

𝑑𝑡2−

𝑘4

𝑘3𝑘7

𝑑(𝑧𝑒𝑗𝑤𝑡)

𝑑𝑡+

𝑘8𝑑(𝑧𝑒𝑗𝑤𝑡)

𝑘3𝑘7𝑑𝑡−

𝑘8𝑘4𝑧𝑒𝑗𝑤𝑡

𝑘3𝑘7 (23)

Taking the derivative of (23),

𝑥0𝑒𝑗𝑤𝑡 =

𝑧𝑗2𝑤2𝑒𝑗𝑤𝑡

𝑘3𝑘7−

𝑘4𝑧𝑗𝑤𝑒𝑗𝑤𝑡

𝑘3𝑘7+

𝑘8𝑧𝑗𝑤𝑒𝑗𝑤𝑡

𝑘3𝑘7−

𝑘8𝑘4𝑧𝑒𝑗𝑤𝑡

𝑘3𝑘7 (24)

Diving (24) by z,

𝑥0

𝑧=

𝑗2𝑤2

𝑘3𝑘7−

𝑘4𝑗𝑤

𝑘3𝑘7+

𝑘8𝑗𝑤

𝑘3𝑘7−

𝑘8𝑘4

𝑘3𝑘7 (25)

Simplifying (25),

𝑥0

𝑧=

𝑗2𝑤2−𝑘4𝑗𝑤+𝑘8𝑗𝑤−𝑘8𝑘4

𝑘3𝑘7

or, 𝑧

𝑥0=

𝑘3𝑘7

𝑗2𝑤2−𝑘4𝑗𝑤+𝑘8𝑗𝑤−𝑘8𝑘4 (26)

If 𝒛

𝒙𝟎= 𝑯𝟎(𝒘) and 𝑯𝟎(𝒘)= |𝑯𝟎(𝒘)|𝒆

𝒋(𝒘𝒕+ɸ(𝒘)), plugging in,

|𝐻0(𝑤)| =𝑘3𝑘7

√(−𝑤2−𝑘4𝑘8)2+((−𝑘4+𝑘8)𝑤)

2 (27)

And

ɸ(𝑤)= −tan−1((−𝑘4+𝑘8)𝑤

−𝑤2−𝑘4𝑘8) (28)

Assuming X0=1, we get

𝑧 =𝑘3𝑘7

√(−𝑤2−𝑘4𝑘8)2+((−𝑘4+𝑘8)𝑤)

2𝑒𝑗(𝑤𝑡−tan−1(

(−𝑘4+𝑘8)𝑤

−𝑤2−𝑘4𝑘8))

(29)

System Response to Unit Impulse

Using (14), we can find our system’s response to unit impulse.

𝑃(𝑡) =𝑘3𝑘7𝑒

𝑘3𝑡

𝑘3+𝑘4−

𝑘3𝑘7𝑒−𝑘8𝑡

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.

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 =𝑘3𝑘7

√(−𝑤2−𝑘4𝑘8)2+((−𝑘4+𝑘8)𝑤)

2𝑒𝑗(𝑤𝑡−tan−1(

(−𝑘4+𝑘8)𝑤

−𝑤2−𝑘4𝑘8))

(29)

The Frequency Response of our system operating under several angular frequencies is shown in

Figure 3a-d.

Figure 2: System Unit Impulse Response

Figure 3a: System Frequency Response, ω = π.

Figure 3b: System Frequency Response, ω = 1.5π.

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.

V. Stability Analysis

Analyzing the stability of the transfer function, from (13)

𝑃(𝑠)

𝐷(𝑠)=

𝑘3𝑘7

(𝑠+𝑘8)(𝑠−𝑘4)=

(𝑘3𝑘7)

𝑠2+(𝑘3−𝑘4)𝑠−𝑘3𝑘4 (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

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).

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.