A Mathematical Modeling of Progression of First Stage of Spontaneous Labor

Patrick Ellsworth, Oberlin College , Dr. Animikh Biswas, University of Maryland, Baltimore County Dr. Rajeshwari Sundaram, Eunice Kennedy Shriver National Institute of Child Health and Human Development

Friedman’s Conjecture The first stage of labor is characterized by dilation of the paturient’s cervix and understanding the rate of this dilation is important for assessing the nearness of delivery and the normalcy of the labor. In the 1950s, obstetrician and researcher Emmanuel Friedman analyzed data from thousands of labors to develop a mathematical model for the rate of dilation, which he conjectured to resemble the curve shown below.

Methods Analysis

Bibliography

I-Splines We use I-splines to model the dynamic process of cervical dilation. I-splines are monotonic spline functions and are used because of the strictly increasing nature of dilation during the first stage of labor. The coefficients for the splines are initially estimated using a least squares estimate.

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Beaver, and William Fraser. "The Natural History of the Normal First Stage of Labor." Obstetrical & Gynecological Survey 65.7 (2010): 414-15. Web.

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3.  Greenberg, Mara B., Yvonne W. Cheng, Margaret Sullivan, Mary E. Norton, Linda M. Hopkins, and Aaron B. Caughey. "Does Length of Labor Vary by Maternal Age?" American Journal of Obstetrics and Gynecology197.4 (2007): n. pag. Web.

4.  Carlhäll, Sara, Karin Källén, and Marie Blomberg. "Maternal Body Mass Index and Duration of Labor." European Journal of Obstetrics & Gynecology and Reproductive Biology 171.1 (2013): 49-53. Web.

Dilation Curves Organizing the data by Parity, BMI, and age, our B-spline models are represented as follows.

Introduction Results

Fig. 1 The 1st stage of labor is divided into 4 hypothetical stages

Goals • We want to assess the rate of change in cervical dilation through the first stage of labor using modern mathematical techniques. • We plan on creating an ODE model of the dilation curve using the paturient’s parity, BMI and age as covariates. • We want to use the dynamical systems framework to assess where a woman actually starts her labor

Our Data: The National Collaborative Perinatal Project

The data we used for our analysis is derived from the National Collaborative Perinatal Project. It includes data from 26,838 paturients who had a singleton term gestation, spontaneous onset of labor, vertex presentation, and a normal perinatal outcome.

Fig 2. A family of I-splines. The linear combination of a basis of I-splines such as these is used to create a smooth curve fit to the data.

An ODE Model Our data also seems well suited for ODE model as we are modeling a time curve. In addition, Friedman’s conjectured sub-stages of the 1st stage of labor suggesting a relation between dilation and its first and second derivatives.

zeuzuuFuFuG γβα +−== )

2(:)(),()(

A Possible Choice for G(u) and an Equation for it’s Optimization Based on

Data

Eq. 1: Let D(t) denote the cervical dilation at time 0.

Plot 1: null parous (black), parity>0(green)

Plot 2: young age(black), average age(green), above average age(red)

Plot 3: normal BMI (black), overweight BMI (green), Obese (red)

The I-spline curves of sub-sections of the data set correspond to other studies2,3,4 on the affects of parity, age and BMI, indicating their validity. However, the overall shape of the curves appears to contradict Friedman’s conjectured shape, which supports the ODE model in Eq 2. that lacks a deceleration phase. Solving Eq 2. using parameters generated from Eq 3, we generate the following plots.

Further Research

Eq. 2

Eq. 3

•  Improve Accuracy •  Establish Covariates