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Simulation model of the relationship between cesarean section rates and labor duration


Cesarean delivery is the most common major abdominal surgery in many parts of the world, and it accounts for nearly one-third of births in the United States. For a patient who requires a C-section, allowing prolonged labor is not recommended because of the increased risk of infection. However, for a patient who is capable of a successful vaginal delivery, performing an unnecessary C-section can have a substantial adverse impact on the patient’s future health. We develop two stochastic simulation models of the delivery process for women in labor; and our objectives are (i) to represent the natural progression of labor and thereby gain insights concerning the duration of labor as it depends on the dilation state for induced, augmented, and spontaneous labors; and (ii) to evaluate the Friedman curve and other labor-progression rules, including their impact on the C-section rate and on the rates of maternal and fetal complications. To use a shifted lognormal distribution for modeling the duration of labor in each dilation state and for each type of labor, we formulate a percentile-matching procedure that requires three estimated quantiles of each distribution as reported in the literature. Based on results generated by both simulation models, we concluded that for singleton births by nulliparous women with no prior complications, labor duration longer than two hours (i.e., the time limit for labor arrest based on the Friedman curve) should be allowed in each dilation state; furthermore, the allowed labor duration should be a function of dilation state.

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Correspondence to Karen T. Hicklin.

Appendix: Percentile matching procedure details

Appendix: Percentile matching procedure details

Since we have estimates of the 5th, 50th, and 95th percentile that are to be matched by the lognormal distribution, we have p1 = 0.05, p2 = 0.50, and p3 = 0.95. So Eq. 4 has the following form

$$\begin{array}{@{}rcl@{}} \frac{\ln (\widehat x_{p_{1}} - \widehat a)-\widehat \lambda}{-\widehat \beta} &= -1.6449 = z_{p_{1}}, \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{\ln (\widehat x_{p_{2}} - \widehat a)-\widehat \lambda}{-\widehat \beta} &= 0 = z_{p_{2}}, \text{ and } \end{array} $$
$$\begin{array}{@{}rcl@{}} \frac{\ln (\widehat x_{p_{3}} - \widehat a)-\widehat \lambda}{-\widehat \beta} &= 1.6449 = z_{p_{3}}. \end{array} $$

From Eq. 7, we obtain

$$ \widehat a = \widehat x_{p_{2}} - \exp (\widehat \lambda). $$

Dividing (6) by Eq. 8, we obtain

$$\frac{\ln (\widehat x_{p_{1}} - \widehat a)-\widehat \lambda}{\rule{0pt}{2.25ex}\ln (\widehat x_{p_{3}} - \widehat a)-\widehat \lambda} = -1, $$

which simplifies to

$$\ln \left[ \left( \widehat x_{p_{1}} - \widehat a \right) \left( \widehat x_{p_{3}} - \widehat a \right) \right] = 2 \widehat \lambda, $$

which we rewrite as

$$ (\widehat x_{p_{1}} - \widehat a)(\widehat x_{p_{3}} - \widehat a)=\exp (2\widehat \lambda). $$

Plugging the right-hand side of Eq. 9 into the left-hand side of Eq. 10 as an “approximation” for \(\widehat a\), then we have the following relation:

$$ \exp (\widehat \lambda) - (\widehat x_{p_{2}} - \widehat x_{p_{1}})][\exp (\widehat \lambda) + (\widehat x_{p_{3}} - \widehat x_{p_{2}})]=\exp (2\widehat \lambda). $$

If we let \(\delta _{21} \equiv \widehat x_{p_{2}} - \widehat x_{p_{1}}\) and \(\delta _{32} \equiv \widehat x_{p_{3}} - \widehat x_{p_{2}}\) where δij presents the difference between quantile estimates for pi and pj, then Eq. 11 simplifies to

$$\exp (2\widehat \lambda) + (\delta_{32}-\delta_{21})\exp (\widehat \lambda) - \delta_{21}\delta_{32} = \exp (2\widehat \lambda), $$

which has the solution

$$\begin{array}{@{}rcl@{}} \widehat \lambda &= &\ln \left( \frac{\delta_{32} \delta_{21}}{\delta_{32}-\delta_{21}}\right) \\ & =& \ln \left[\frac{(\widehat x_{p_{3}} - \widehat x_{p_{2}})(\widehat x_{p_{2}} - \widehat x_{p_{1}})}{\widehat x_{p_{3}}-2 \widehat x_{p_{2}} + \widehat x_{p_{1}}}\right]. \end{array} $$

Using this value of \(\widehat \lambda \), we evaluate \(\widehat a\) as follows:

$$ \widehat a = \max\left\{\widehat x_{p_{2}} - \exp(\widehat \lambda), 0 \right\}. $$

This modification of Eq. 9 is needed because we must have \(\widehat a \ge 0\) for a nonnegative shifted lognormal random variable X. Finally, we evaluate \(\widehat \beta \) by plugging the values of \(\widehat \lambda \) and \(\widehat a\) into Eq. 8, yielding

$$ \widehat \beta = \frac{\ln (\widehat x_{p_{3}}-\widehat a)-\widehat \lambda}{z_{p_{3}}}; $$

then the associated lognormal distribution has mean

$$\widehat \mu = \widehat a + \exp\left\{\widehat \lambda +\frac{1}{2}\widehat \beta^{2} \right\} $$

and standard deviation

$$\widehat \sigma = \widehat \mu \sqrt{\exp(\widehat \beta^{2}) - 1}. $$

If \(\widehat a = 0\) in Eq. 13, then the correct formulas for \(\widehat \lambda \) and \(\widehat \beta \) are

$$ \widehat \lambda = \ln \left( \widehat x_{p_{2}}\right) $$


$$ \widehat \beta = \frac{\ln \left( \widehat x_{p_{3}}\right)-\ln\left( \widehat x_{p_{2}} \right)}{z_{p_{3}}}. $$

Note carefully that Eq. 15 is not the same as Eqs. 12, and 16 is not the same as Eq. 14.

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Hicklin, K.T., Ivy, J.S., Wilson, J.R. et al. Simulation model of the relationship between cesarean section rates and labor duration. Health Care Manag Sci 22, 635–657 (2019).

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  • Medical decision making
  • Mode of delivery
  • Birth
  • Simulation
  • Dystocia
  • Percentile matching