^{1}Although many physical and physiologic factors have been linked to the amount of labor pain women experience,

^{2}it is difficult to discern the contribution of individual factors because the pain intensity changes with cervical dilation.

^{1}As such, a model of labor progress is required to assess the effects of patient demographics and medical interventions on labor pain.

^{3,4}Recently, studies have shown that ethnic differences exist in the duration of the second stage of labor and in the likelihood of Cesarean delivery.

^{5,6}Even though the use of ethnicity as a marker of genetic variation has been debated,

^{7}we hypothesized that a patient's self-identified ethnicity might predict labor pain and progress.

#### Materials and Methods

*e.g.*, when membranes were ruptured) and labor interventions (

*e.g.*, when labor was induced, when oxytocin was administered, when analgesia was initiated) from the electronic medical record. We also obtained maternal weight, maternal height, maternal age, gestational age, and birth weight that were recorded from a structured nursing interview from the electronic medical record. Self-reported ethnicity was recorded by labor and delivery administrative personnel according to categories derived from the 2000 census. Although people of Hispanic ethnicity can be of any race, patients are asked to choose only one category.

##### Statistics and Data Analysis

*P*< 0.05 was considered significant. The labor progress and labor pain models were analyzed with NONMEM (Nonlinear Mixed-Effects Modeling; Globomax, Ellicott City, MD) using PLT Tools (PLT Soft, San Francisco, CA). One hundred parturients were enrolled in each group; in our previous study with 100 subjects in test and training sets, we were able to identify a covariate effect with the magnitude of 1 NRS point.

^{1}We consider this difference to be minimally clinically significant.

##### Labor Progress Model

^{8}produced fits that were extraordinarily dependent on initial parameter estimates and not suitable for hypothesis testing. We tested linear, sigmoidal, and single exponential functions that were significantly inferior to a biexponential model (see appendix).

^{−λ1t}+(10−C)e

^{−λ2t}, where λ

_{1}and λ

_{2}are rate constants of active and latent labor, respectively, and where C is the number of centimeters of dilation associated with the active phase of labor. τ is time. This model produces similar predictions to the bilinear model, but it is suitable for hypothesis testing with NONMEM. We used an exponential model for interindividual variability and an additive model for intraindividual variability. The full data set is provided as an Excel file (see Supplemental Digital Content 1, http://links.lww.com/ALN/A556), and the NONMEM control files (see Supplemental Digital Content 2, http://links.lww.com/ALN/A557, which is the NONMEM control file for the final labor progress model, and Supplemental Digital Content 3, http://links.lww.com/ALN/A558, which is the NONMEM control file for the final labor pain model) are available as text files.

_{1}, λ

_{2}, or C. For example, λ

_{1}= θ

_{1}+ θ

_{2}× (patient age − median age), where θ

_{1}is the nominal value of the parameter and θ

_{2}is effect of age on the parameter. We modeled the effect of ethnicity by introducing new parameters into the model for each ethnic group. We modeled the effect of abrupt changes in patient state during labor (initiation of neuraxial analgesia, rupture of membranes, initiation of an oxytocin infusion) by using a time scale factor, as described in the appendix.

*P*< 0.01 with 1 degree of freedom). The model parameters were estimated by NONMEM using the “first order conditional estimation” approach. We chose

*P*< 0.01 rather than

*P*< 0.05 in building the models to compensate (in part) for the large number of models examined in the process of model development.

*P*< 0.05, 1 degree of freedom). The bootstrap was calculated by randomly sampling a new data set from the patient data, with replacement, and then repeating the NONMEM analysis. This procedure was repeated 1,000 times. A 95% confidence interval was calculated from the bootstrap analysis as the interval from the parameter value at the 2.5% rank to the parameter value at the 97.5% rank. The log likelihood profile addresses the confidence in the parameter relative to the overall model. The bootstrap analysis addresses the confidence in the parameter relative to the data.

##### Labor Pain Model

^{1}The NRS pain response to cervical dilation was fit to a sigmoid equation:

_{MIN}is the minimum reported pain score, NRS

_{MAX}is the maximal reported pain score, CD

_{50}is the cervical dilation associated with 50% of maximal pain, and γ is the steepness of the sigmoidal relationship. This model assumes that for a population of laboring women, reported pain with contractions is modest when the cervix is closed and increases in intensity as the cervix dilates, reaching maximum intensity when the cervix is fully dilated. The assumption of monotonicity was verified by tallying the number of women who reported a decrease in pain during labor and by qualitatively assessing the distribution of bias and the fit of the model to the raw data.

*post hoc*individual Bayesian predicted cervical dilation from the labor progress model, rather than the actual measured dilation. First, some pain measurements did not have associated cervical dilations. Using the individual

*post hoc*Bayesian predicted dilations provided a predicted cervical dilation at every point in time. Second, the

*post hoc*Bayesian predicted dilation is arguably a more accurate measurement than an actual measurement, both because the actual measurement is always reported as an integer and because the prediction represents, in part, an average of many predictions. Consider progressive cervical examinations of 4, 4, 5, and 5. A more accurate estimate of the “true” cervical dilation might be 4.0, 4.5, 5.0, and 5.5 cm, which is what the

*post hoc*Bayesian prediction would reflect.

*rate*of dilation on each of the model parameters.

_{MIN}and NRS

_{MAX}, and we used an exponential model for the interindividual variability in CD

_{50}. Because interindividual variability was miniscule on NRS

_{MAX}, it was fixed in our final model. No interindividual variability was modeled for γ. We used an additive model of intraindividual residual error. Control files for the final model is provided as Supplemental Digital Content 3, http://links.lww.com/ALN/A558.

_{50}. This hindered NONMEM from accurately searching the parameter space. After verifying that the steepness of the relationship around CD

_{50}had not precluded NONMEM from converging on the optimal model, log likelihood profiles were run approximately 100 times with different starting parameters and occasionally with some parameters of the model fixed to verify that the curves were unimodal and to force NONMEM to further explore the parameter space. Lastly, for several log likelihood profiles, some residual jaggedness was removed by manually selecting local minimum values.

#### Results

Fig. 1 Image Tools |
Table 1 Image Tools |

##### Labor Progress Model

*P*values shown in table 3 represent the

*P*values for the incremental improvement, and they are thus conservative relative to the improvement that might be observed from assessment of the covariate effect in the absence of other explanatory covariates.

Fig. 2 Image Tools |
Fig. 3 Image Tools |

*post hoc*predictions (gray lines). Figures 2C and 2D show the error in the fits, based on the population values of the parameters (

*e.g.*, with interindividual variability set to 0). The fit for the initial and final models are almost visually indistinguishable. Indeed, the median absolute prediction error for the initial model is 0.98 cm, which the model incorporating covariates has only improved to 0.94 cm. Figure 3 shows the predicted (X axis)

*versus*measured (Y axis) cervical dilation for (A) the initial model and (B) final covariate adjusted model. As noted for figure 2, the improvement in fit with the inclusion of covariates is almost invisible in the figure. Thus, even though the covariate effects shown in table 3 are highly statistically significant, they represent very small improvements in the model.

Fig. 4 Image Tools |
Fig. 5 Image Tools |

_{1}), latent labor rate constant (λ

_{2}), and the coefficient (C) on the model parameters in the Non-Asian and Asian patients. Figure 5 shows log-likelihood profiles and 95% confidence intervals for the neuraxial analgesia scale factor and the effect of weight on the active and latent rate constants. In every case, the parameter estimates are well constrained within the boundaries of the model and do not cross 0, showing that they are statistically different from 0. The likelihood profiles similarly reach the edges of the 95% confidence interval derived from the bootstrap analysis at roughly the

*P*< 0.01 range. This shows that the likelihood profiles are not dependent on particular patients or observations.

*post hoc*Bayesian fits for the model used to interpolate cervical dilations. The only covariate incorporated in this model is the presence of neuraxial analgesia. As explained in the methods, labor pain was modeled as a function of the cervical dilation predicted by the labor progress model. This provided a predicted cervical dilation at every timepoint. The model was based on incorporation of the neuraxial analgesia effect, which is associated with a different progress of labor. The median error was 0.01 cm and the median absolute error was 0.42 cm. Figure 7B shows the final

*post hoc*Bayesian estimate for the final covariate model. It is almost indistinguishable visually from the top graph. This model was not used for the labor pain analysis because it excludes those patients whose weights were not recorded. The bias and accuracy is similar to the upper figure, with a median error of +0.005 cm and a median absolute error of 0.43 cm.

*post hoc*Bayesian model (fig. 7B, median error = 0.43 cm). There is considerable intersubject variability that is not accounted for by the covariates assessed in this study.

##### Labor Pain Model

*P*values shown in table 4 represent the incremental improvement, first of incorporating self-identified ethnicity, and then from incorporating dilation rate.

*post hoc*as an outlier.

_{MAX}profile. These proved to be an artifact of NONMEM's inability to fully explore the parameter space, were addressed by calculating log likelihood profiles dozens of times, and selecting minimum values that represented the true minimum (or, at least, as close to the true minimum as NONMEM could identify). The 95% confidence intervals from the bootstrap analysis agree well with the log likelihood profiles, with the exception of the rate scalar on γ. The lower bound of the bootstrap analysis was very close to 0, which reflects the dependence on that parameter on a single point, circled in figure 8. Indeed, this is a demonstration of how the bootstrap shows robustness of a parameter estimate relative to the data, whereas the log likelihood profile shows the robustness of the parameter estimate relative to the model.

*post hoc*Bayesian individual pain model. The dashed line snaking through the center of the figure is a supersmoother (a moving average,

^{9}implemented in the R programming language). A bias is evident in the population fit (upper graph) at cervical dilations less than 3 cm. This is related to the prediction of the “typical” baseline NRS pain score of approximately 1, which results in an asymmetric distribution of the prediction error; patients can have pain that is 9 points higher than the prediction for a typical individual, but only one point lower than the prediction for a typical individual. Once the cervix has dilated to 3 cm, the model predicts a pain score of 5 in the typical individual, and error in the prediction becomes symmetrical. The bias is not seen in the predicted pain scores for individual patients (b) because the model for each individual adjusts the predicted pain score when the cervix is not dilated to the actual level of pain when the patient entered the study. The median absolute residual error was 1.7 NRS points for the population model, and 0.95 NRS points for the individual

*post hoc*Bayesian model. The difference between the models shows the magnitude of the still unexplained intersubject variability.

#### Discussion

^{1}permitted testing the influence of the patient covariates self-identified ethnicity, maternal age, maternal weight, maternal height, birth weight, and gestational age on labor progress and pain. Examining the labors of 500 consecutive nulliparous parturients from 5 self-identified ethnicities, we found that ethnicity and other demographic variables impart highly significant differences in labor progress and labor pain. However, the magnitude of the effects of patient covariates is quite small and is unlikely to have great clinical significance. Our findings do not suggest that targeting a specific self-identified ethnicity is important to reduce variability in future prospective trials of labor pain.

^{10}In contrast to our findings, a large retrospective cohort study from University of California, San Francisco, did not identify a difference in the first stage of labor according to ethnicity. However, their methods differed from ours in that they did not study the latent and active labor separately; they only considered the length of the first and second stages of labor together.

^{5}Our findings demonstrate the enhanced model sensitivity imparted by considering each cervical exam rather than just the total time before delivery. Potentially consistent with our findings, Greenberg

*et al.*found that both nulliparous and multiparous Black women had a shorter second stage of labor when compared with White women, whereas Asian women experience a longer second stage of labor and higher Cesarean section rates.

^{5,11}It is possible that the slower rate of labor in Asians in the active phase may contribute to Cesarean sections for failure to progress. All subjects screened for this study had a vaginal delivery; as such, we do not know whether our Asian and White patients had a higher Cesarean section rate. It is likely that our results are biased to some extent by limiting our population to women who delivered their children vaginally.

^{12}Several authors have documented this historical change.

^{13,14}Population demographics and the medical management of labor have changed markedly in the years since Friedman reported his results. Particularly, the induction of labor is common, and use of oxytocin has increased and represented 27% and 60% of our sample respectively. While labor induction and oxytocin use did not vary according to ethnicity, it might account for the overall slower progress of labor compared to historical data. The mothers in our study are older and heavier than previously.

^{15,16}In our sample, high maternal weight was predictive of slower labor (fig. 6). This association has been noted previously and may contribute to a higher risk of Cesarean delivery in obese women.

^{15,16}There may be a direct effect of obesity on the myometrium. Myometrium from obese women taken at Cesarean section contracted with less force and frequency and had reduced calcium transients when compared with uterine tissue from lean women.

^{17}

^{18}This has typically not been validated in prospective, randomized controlled studies, which suggests that the relationship between neuraxial analgesia and labor slowing is not causal.

^{19–23}In addition, studies have correlated early severe labor pain with dystocia

^{24–26}; as a result, a temporal association of neuraxial anesthesia with slowed labor is an expected finding.

*P*= 0.011 and

*P*= 0.024, respectively. Thus, patients with slower labor (smaller rate constants) received earlier neuraxial analgesia. These data are consistent with, but do not prove, the association of neuraxial analgesia placement with slower labor progress is not causal.

^{6}However, we did not observe an association between birth weight and the rate of labor progress.

^{1,6}Instantaneous labor rate was a significant covariate for labor pain in our study. Although logical and apparently valid, the statistical significance of this finding was dependent on the data from a single Asian patient who had a slow labor and little pain and as such should be taken with caution.

^{27,28}Previous studies have shown differences among ethnic groups with respect to stoicism, healthcare quality, and pain coping strategies. Prepared childbirth and breathing techniques have been shown to reduced pain scores in early labor.

^{2,29}We do not know which patients had professional childbirth preparation; as such, we cannot determine whether this variable contributed to the difference that we have identified.

_{MAX}. There was little variability in NRS

_{MAX}. All subjects without analgesia reported significant pain close to full dilation. As such, in our final model we fixed intraindividual variability at 0 on this term. Similarly, it is possible that women having a very rapid painful labor would be more likely to be rushed to a labor room and have an epidural placed before the nurse could take a pain score with their admission exam. As such, pain might be underrepresented. The impact of the frequent use of epidural analgesia in our population can be evaluated with a similar study in a population with lower use of epidural analgesia. Similarly, the interpretation of our data are affected by informed censoring, in that patients with higher levels of pain are more likely to receive epidural analgesia and have censored observations. Fortunately, a mixed effects approach is fairly robust, even in the presence of informed censoring.

^{30}

#### Appendix: Population Model of Labor Progress

^{8,12}which was later simplified to a model of two straight lines. The first line is used to describe latent labor, were the cervix dilates at approximately 0.2–0.5 cm/h. The second line describes active labor, where the cervix dilates at 1–2 cm/h. Traditionally, women have been considered to transition from latent to active labor when the cervix is dilated to approximately 4–5 cm.

_{Latent}, the rate (slope) of latent labor, M

_{Active}, the rate (slope) of active labor, and Inflection, the cervical dilation where women transition from latent labor to active labor. The time of the transition from latent to active labor (

*i.e.*, the time when the curves intersect) can be readily calculated as (10 − Inflection)/M

_{Active}. The bilinear model is necessarily bounded at 0 and 10, as cervical dilation cannot be negative, and full dilation is defined as 10 cm.

Fig. 14 Image Tools |
Fig. 15 Image Tools |

*post hoc*step. The population fit of the bilinear model to all of the data are shown in figure 14. There is a considerable amount of variability about the typical prediction, with a median absolute error, |(measured dilation − predicted dilation)|, of 1.0 cm. The goodness of fit is shown in figure 15 for the population estimate (A) and the individual

*post hoc*Bayesian estimates (B). The −2 log likelihood function for the bilinear model is 2741.

*i.e.*, at the transition from latent to active labor). Consider an observation immediately to the left of the transition point. A change in either slope, or in Inflection, might move the transition point so that the observation is now to the right of the transition. Suddenly, this data point is being fit to an entirely different line, with unpredictable results. The lack of a smooth transition (or, more precisely, the lack of a derivative at the transition point) makes the model profoundly sensitive to starting estimates. This profound sensitivity to starting estimates preclude use of the model for hypothesis testing. The bounding of the model at 0 may also contribute to the instability of the model, as the derivative is undefined for all points to the right of the intersection of the line for latent labor with the X axis.

_{0}is the cervical dilation at time 0 (typically 10 by definition), CD

_{min}is the minimal cervical dilation (which should be 0, but because women arrive at the hospital somewhat dilated was 0.43 cm in the final model), Time is the number of hours before full dilation, Time

_{50}is the time of half dilation, and γ is the steepness of the transition. The model follows the data better than the simple bilinear model, as suggested by the −2 log likelihood of 2,499, a decrease of 242 points from the log likelihood for the bilinear model. The sigmoidal model provides for a terminal deceleration in labor progress, which is not evident in the raw data, is virtually invisible in the function, and has been disputed by several investigators.

^{13,31}

^{9}implemented in the R programming language) applied to the data. The best bilinear fit appears as the dashed line. The supersmoother curve suggests a biexponential model, which we implemented with the function Dilation = Ce

^{−λ}

_{1}

^{t}+ (10−C)e

^{−λ}

_{2}

^{t}. This function has the same number of parameters as the model with two intersecting lines. λ

_{1}and λ

_{2}are rate constants, with units of inverse time. During latent labor, cervical dilation will double every 0.693/λ

_{2}hours. During active labor, cervical dilation will double approximately 0.693/λ

_{1}hours. C can be thought of as approximately the number of centimeters of dilation associated with the active phase of labor. Therefore, 10-C is approximately the number of centimeters of dilation associated with the latent phase of labor, and represents the transition point between latent and active labor. The transition from latent to active labor occurs gradually; therefore, it is probably best to not focus on the value of C or the specific exponents, but rather to consider the function in its entirety. The rate of cervical dilation is the derivative of Dilation = Ce

^{−λ}

_{1}

^{t}+(10−C)e

^{−λ}

_{2}

^{t}, with respect to time, which can be calculated as Rate (cm/h) = Cλ

_{1}e

^{−λ}

_{1}

^{t}+ (10−C)λ

_{2}e

^{−λ}

_{2}

^{t}.

^{−λ1t}, This curve produced a −2 log likelihood that was 178 points worse than the biexponential model, and was not further considered.

*P*= 0.05 and

*P*= 0.01.

*i.e.*, each bootstrap analysis considers a resampled population, not the same data set as used in calculating the −2 log likelihood profile), it is an expected result that the 95% confidence intervals from the bootstrap analysis do not exactly match the intervals suggested by the −2 log likelihood profiles.

^{−λ 1t}+ (10−C)e

^{−λ1t}can be calculated as the derivative with respect to time: Rate(cm/h) = λ

_{1}Ce

^{−λ1t}+ λ

_{2}(10−C)e

^{−λ1t}. Figure 21 shows the rate of cervical dilation per hour (A) and the rate of dilation as a function of cervical dilation (B).

_{1}, λ

_{2}, and C.

_{1}, λ

_{2}, and C; that approach would produce two distinct curves, with an abrupt discontinuity in predicted cervical dilation at the moment of transition. We could have used a mathematical function to constrain the model to a smooth function at the moment of a time-varying covariate, but we could find no precedent for that approach nor an intuitively tractable mathematical model. Instead, we adopted a very simple approach that ensured a smooth transition. We introduced a “time adjustment” factor that accelerated (or decelerated) the time axis at the transition moment. This required introducing only a single term, which represented how the time-varying covariate accelerated (factor > 1) or decelerated (factor < 1) the progress of labor. The factor, θ, was defined as the time interval from the moment of the intervention until 10 cm of dilation in the absence of the intervention divided by the time interval from the moment of the intervention until 10 cm of dilation in the presence of the intervention. To express it more formally, let T

_{Observed}be the actual time between the intervention and full dilation (which is the same as the time of the intervention) and T

_{NoIntervention}be expected time if the person had not had the intervention. We define the time scale factor, θ, as:

_{Observed}is greater than T

_{NoIntervention}, and labor is slowed. For example, θ = 0.6, then labor runs at only 60% of the expected pace after an intervention. If θ > 1, then T

_{Observed}is less than than T

_{NoIntervention}, meaning that labor is accelerated by the intervention. For example, if θ = 2, then the pace of labor is doubled following the intervention.

_{unscaled}, which is the time recorded in the study, and t

_{scaled}, which is a scaled time that has expanded or contracted from the intervention, and is the value of time used in the labor progress model. The calculation of t

_{scaled}can be derived from the definition of θ.

_{scaled}must include the full impact of the intervention on the time course. Based on the definition of θ, T

_{Nolntervention}= θ × T

_{Observed}. The full impact is the difference between the observed time, T

_{Observed}, and the expected time in the absence of the intervention, θ × T

_{Observed}. Thus, t

_{scaled}= t

_{unscaled}+ θ × T

_{Observed}− T

_{Observed}= t

_{unscaled}+ T

_{Observed}(θ−1).

_{scaled}= T

_{NoIntervention}, and t

_{unscaled}= T

_{Observed}. By the definition of θ T

_{Nolntervention}= θ × T

_{Observed}; therefore, at the moment of the intervention t

_{scaled}= t

_{unscaled}× θ. This relationship holds after the intervention as well: t

_{scaled}= t

_{unscaled}×θ.