RESEARCH ARTICLE

A compelling symmetry: The extended

fetuses-at-risk perspective on modal, optimal

and relative birthweight and gestational age

K. S. Joseph

ID

*

Department of Obstetrics and Gynaecology, School of Population and Public Health, University of British

Columbia and the Children’s and Women’s Hospital and Health Centre of British Columbia, Kelowna, Canada

* kjoseph@cw.bc.ca

Abstract

Background

The relationship between several intriguing perinatal phenomena, namely, modal, optimal,

and relative birthweight and gestational age, remains poorly understood, especially the

mechanism by which relative birthweight and gestational age resolve the paradox of inter-

secting perinatal mortality curves.

Methods

Birthweight and gestational age distributions and birthweight- and gestational age-specific peri-

natal death rates of low- and high-risk cohorts in the United States, 2004–2015, were estimated

using births-based and extended fetuses-at-risk formulations. The relationships between these

births-based distributions and rates, and the first derivatives of fetuses-at-risk birth and perina-

tal death rates were examined in order to assess how the rate of change in fetuses-at-risk

rates affects gestational age distributions and births-based perinatal death rate patterns.

Results

Modal gestational age typically exceeded optimal gestational age because both were influ-

enced by the peak in the first derivative of the birth rate, while optimal gestational age was

additionally influenced by the point at which the first derivative of the fetuses-at-risk perinatal

death rate showed a sharp increase in late gestation. The clustering and correlation

between modal and optimal gestational age within cohorts, the higher perinatal death rate at

optimal gestational age among higher-risk cohorts, and the symmetric left-shift in births-

based gestational age-specific perinatal death rates in higher-risk cohorts explained how rel-

ative gestational age resolved the paradox of intersecting perinatal mortality curves.

Conclusions

Changes in the first derivative of the fetuses-at-risk birth and perinatal death rates underlie

several births-based perinatal phenomena and this explanation further unifies the fetuses-

at-risk and births-based models of perinatal death.

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OPEN ACCESS

Citation: Joseph KS (2020) A compelling

symmetry: The extended fetuses-at-risk

perspective on modal, optimal and relative

birthweight and gestational age. PLoS ONE 15(11):

e0238673. https://doi.org/10.1371/journal.

pone.0238673

Editor: Frank T. Spradley, University of Mississippi

Medical Center, UNITED STATES

Received: August 19, 2020

Accepted: November 12, 2020

Published: November 30, 2020

Peer Review History: PLOS recognizes the

benefits of transparency in the peer review

process; therefore, we enable the publication of

all of the content of peer review and author

responses alongside final, published articles. The

editorial history of this article is available here:

https://doi.org/10.1371/journal.pone.0238673

Copyright: © 2020 K. S. Joseph. This is an open

access article distributed under the terms of the

Creative Commons Attribution License, which

permits unrestricted use, distribution, and

reproduction in any medium, provided the original

author and source are credited.

Data Availability Statement: The data are publicly

available at https://www.cdc.gov/nchs/data_

access/vitalstatsonline.htm.

Introduction

Several studies have shown that population cohorts based on nationality, racial origin and

other characteristics vary substantially in terms of birthweight distribution and optimal birth-

weight (i.e., the birthweight at which perinatal mortality rates are lowest) [1–9]. A related enig-

matic finding is that optimal birthweight typically exceeds modal birthweight (i.e., the

maximum of the birthweight distribution) [7–9]. Although it is unclear why many fetuses in

diverse populations are born before reaching optimal size, these findings have led to recom-

mendations regarding the need for population-specific standards of birthweight for identifying

small infants at risk of perinatal death [8].

Some support for the proposition that perinatal mortality risk is best assessed through pop-

ulation-specific standards of birthweight is also forthcoming from the literature on the para-

dox of intersecting perinatal mortality curves. This phenomenon was first described over 50

years ago by Yerushalmy who showed that neonatal death rates favoured the low birthweight

infants of mothers who smoked (compared with the low birthweight infants of mothers who

did not smoke), while the opposite was true at higher birthweights [10]. The paradox is now

recognized to be a general phenomenon [11–25] that is observed across numerous contrasts

(e.g., infants of hypertensive vs normotensive mothers [14], and singletons vs twins

[13,15,16]), outcomes (e.g., stillbirths and cerebral palsy [11–19]) and indices of prematurity

(gestational age and birthweight [11–25]). One of the first attempts at resolving the paradox

involved an intriguing reformulation involving relative birthweight and relative gestational

age (i.e., with absolute birthweight or gestational age in each population recast in terms of its

mean and standard deviation) [7,17]. When birthweight- and gestational age-specific perinatal

death rates are quantified in terms of relative birthweight or relative gestational age, infants of

mothers who smoke (have hypertension, etc) have higher rates of perinatal death at all birth-

weights and gestational ages [5–7,9,12,14,15,17,25–28].

A recent paper [29] offered evidence in favour of the proposition that the rate of change in

the birth rate of a population (i.e., the first derivative of the population’s fetuses-at-risk birth

rate) determines the population’s gestation age distribution, and that the first derivatives of the

birth rate and the fetuses-at-risk perinatal mortality rate together determine the population’s

births-based gestational age-specific perinatal mortality pattern. This unifies the fetuses-at-risk

and births-based models of perinatal death and also explains various perinatal phenomena

including the early gestation exponential decline and the late gestation exponential increase in

births-based perinatal mortality rates, and also the paradox of intersecting perinatal morality

curves [29,30]. In this paper, the first derivatives of the birth rate and the fetuses-at-risk perina-

tal mortality rate are used to explain other previously unexplained phenomena, namely,

modal, optimal and relative birthweight and gestational age. Understanding these phenomena,

especially the mechanism by which relative gestational age uncrosses intersecting perinatal

mortality curves, will provide further support for unifying the two models of perinatal death.

Methods

Background and rationale for the study

The seemingly opposed perspectives of the births-based and fetuses-at-risk models [29] can be

reconciled by viewing the early gestation exponential decline in births-based perinatal death

rates as being the product of an initially accelerating birth rate (i.e., steep increase in the first

derivative of the fetuses-at-risk birth rate) and a fetuses-at-risk perinatal death rate that is sta-

ble or slowly accelerating in early gestation (no change or a small increase in the first derivative

of the fetuses-at-risk perinatal death rate). Similarly, the late gestation increase in births-based

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Funding: KSJ’s work is supported by an

Investigator award from the BC Children’s Hospital

Research Institute. The funder had no role in study

design, data collection and analysis, decision to

publish, or preparation of the manuscript.

Competing interests: The author has declared that

no competing interests exist.

perinatal death rates can be explained as a product of a decelerating birth rate (i.e., sharp

declines in the first derivative) and an abrupt acceleration in the fetuses-at-risk perinatal death

rate (i.e., sharp increase in the first derivative). Births-based perinatal death rates fall exponen-

tially in early gestation because the accelerating birth rate results in an increasing number of

births, whereas the number of perinatal deaths is essentially unchanged as a consequence of

the stable or slowly accelerating fetuses-at-risk perinatal death rate. On the other hand, the late

gestation rise in births-based perinatal death rates occurs because reductions in acceleration

(or a deceleration) in the birth rate at later gestation leads to a relatively smaller increase (or a

fall) in the number of births, whereas the number of perinatal deaths rises sharply because of

the rapidly accelerating fetuses-at-risk perinatal death rate [29,30]. Compared with low-risk

cohorts, higher-risk cohorts show a steeper increase in the first derivative of the birth rate at

early gestation (i.e., greater acceleration in the birth rate), and an earlier peak and an earlier

decline in this first derivative at late gestation (i.e., earlier reductions in acceleration in the

birth rate). The left-shift in the distribution of the first derivative of the birth rate in higher-

risk cohorts is responsible for a left-shift in gestational age distributions and in births-based

perinatal death rate curves. The latter left-shift in births-based perinatal death rates of higher-

risk cohorts results in the paradox of intersecting perinatal mortality curves [29,30].

The rationale for the present study was premised on the above-mentioned propositions: if

the rate of change in the birth rate determines the birth rate pattern and influences the gesta-

tional age distribution, and if the rate of change in the birth rate and the rate of change in the

fetuses-at-risk perinatal death rate together influence the pattern of births-based gestational

age- and birthweight-specific perinatal death rates, it is likely that the rate of change in fetuses-

at-risk birth and perinatal death rates also underlie the phenomena of modal, optimal, and rel-

ative birthweight and relative gestational age. The rate of change in the birth rate is of particu-

lar interest as it’s magnitude at specific points in gestation is not immediately evident from the

exponentially increasing birth rate.

Data source and analysis

All live births and stillbirths in the United States from 2004 to 2015 were included in the study

with data obtained from the fetal death and period linked birth-infant death files of the

National Center for Health Statistics. The study population was restricted to births with a clini-

cal estimate of gestation between 20 and 43 weeks. Twelve low- and high-risk cohorts were

identified, namely, singletons of women who did not have hypertension or diabetes (referred

to as low-risk singletons), singletons of women with hypertension, singletons of women with

diabetes, singletons of women with hypertension and diabetes, White singletons, Black single-

tons, singletons of women aged 25–29 years, singletons of women aged �35 years, singletons

of women with a previous preterm birth, singletons of women without a previous preterm

birth, twins, and triplets.

Preliminary examination of the birthweight distribution showed substantial ounce and

digit preference in birthweight values (S1 Fig in S1 Appendix) and birthweight was therefore

categorized into 28 g birthweight groups centred on the gram equivalent of each complete

ounce. The birthweight distribution and its modal value, and the birthweight-specific perinatal

death rate (including stillbirths and neonatal deaths) and its lowest point (i.e., optimal birth-

weight) were then estimated by fitting splines to the log transformed birthweight groups and

birthweight-specific perinatal death rates using the Transreg procedure in the SAS statistical

software package (SAS Institute, Cary, NC).

The frequency distribution of gestational age and gestational age-specific perinatal death

rates were calculated under the births-based formulation (expressed per 1,000 total births) and

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modal and optimal gestational age were estimated. Gestational age-specific birth rates and ges-

tational age-specific fetuses-at-risk perinatal death rates (both expressed per 1,000 fetus-

weeks) were also calculated using the extended fetuses-at-risk formulation [28,31–36]. The

number of births (or perinatal deaths) at any gestational week constituted the numerator for

these fetuses-at-risk rates, while the fetal-time accrued by the fetuses at risk over the gestational

week in question constituted the denominator. Fetal-time was estimated by averaging the

number of fetuses at the beginning and the end of the gestational week of interest (which

included fetuses delivered at that gestational week and those delivered subsequently; S1 and S2

Tables in S1 Appendix).

The Expand procedure in the SAS statistical package was used to estimate the first deriva-

tives of the fetuses-at-risk gestational age-specific birth rates and the fetuses-at-risk gestational

age-specific perinatal death rates (S3 Table in S1 Appendix). The first derivatives were com-

puted from cubic splines fitted to the fetuses-at-risk birth and perinatal death rates and quanti-

fied the rate of change (increase or decrease) in these rates at each gestational week. It may be

helpful to view the birth rate (births per 1,000 fetus-weeks) and its first derivative (births per

1,000 fetus-weeks per week, or births per 1,000 fetus-weeks

2

) as being analogous to velocity

(metres/sec) and acceleration/deceleration (metres per second per second, or metres per sec-

ond

2

), respectively. Thus, a positive first derivative of the birth rate represents an accelerating

birth rate while a negative first derivative represents a decelerating birth rate. A positive and

continually increasing first derivative of the birth rate signifies a progressively increasing accel-

eration in the birth rate, while a positive and progressively decreasing first derivative signifies a

birth rate that is increasing but at a slower rate (i.e., with reduced acceleration) than in previ-

ous gestational weeks.

Birthweight and gestational age distributions, gestational age-specific birth rates, the deriva-

tives of the birth rates, births-based and fetuses-at-risk perinatal death rates, and the deriva-

tives of the fetuses-at-risk perinatal death rates were estimated for each low- and high-risk

cohort and graphed in order to examine potential relationships with modal, optimal and rela-

tive birthweight and gestational age (i.e., with the latter calculated using z-scores based on the

mean and standard deviation of the birthweight and gestational age distributions of each

cohort). Correlations between the gestational age at which the first derivative of the birth rate

peaked and the mean, mode, median and optimal birthweight and gestational age were esti-

mated in the 12 cohorts using Pearson correlation coefficients (r). Correlations between the

gestational age at which the first derivative of the fetuses-at-risk perinatal death rate showed an

abrupt upward increase at late gestation and optimal birthweight and optimal gestational age

were similarly assessed.

All analyses were based on anonymized, publicly available data and ethics approval for the

study was not sought.

Results

There were 47,626,172 live births and stillbirths between 20 and 43 weeks’ gestation in the

study population. The rate of perinatal death varied substantially between the different cohorts;

it was 8.2 per 1,000 total births among low-risk singletons, and 72.4 per 1,000 total births

among triplets (S4 Table in S1 Appendix).

Fig 1A and 1B shows birthweight distributions, birthweight-specific perinatal death rates

and modal and optimal birthweight among low-risk singletons and twins. Modal birthweight

was substantially lower than optimal birthweight in both cohorts, and modal birthweight and

optimal birthweight were substantially lower among twins; similarly, modal and optimal gesta-

tional age were lower among twins (37 and 38 weeks, respectively) compared with low-risk

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singletons (39 weeks and 40 weeks, respectively; Fig 1C and 1D). The lowest gestational age-

specific perinatal death rate among twins was higher than the lowest perinatal death rate

among low-risk singletons. The births-based perinatal death rate curves of the two cohorts

intersected; perinatal death rates were lower among twins <38 weeks’ and higher at 38 weeks’

gestation and over compared with perinatal death rates among low-risk singletons (Fig 1E).

When gestational age-specific perinatal death rates were based on relative gestational age (z-

scores), twins had higher rates of perinatal death at all gestational ages (Fig 1F).

Fig 2 shows the birth rate, the rate of change in the birth rate and the gestational age distri-

bution among the singletons of low-risk women and twins. The first derivative of the birth rate

was left-shifted (Fig 2B), the birth rate was considerably higher at each gestational week (Fig

2A), and the gestational age distribution was substantially left-shifted among twins (Fig 2C).

Fig 3 shows the birth rates and their first derivatives, the fetuses-at-risk perinatal death rates

and their derivatives and births-based perinatal death rates in the two cohorts. The first

Fig 1. Birthweight distributions and birthweight-specific perinatal death rates among singletons of low-risk women (i.e., without

hypertension or diabetes; Panel A) and twins (Panel B); gestational age distributions and gestational age-specific perinatal death rates among

singletons of low-risk women (Panel C) and twins (Panel D); and births-based gestational age-specific perinatal death rates (Panel E) and

births-based relative gestational age-specific perinatal death rates (Panel F) among singletons of low-risk women and twins, United States,

2004–2015.

https://doi.org/10.1371/journal.pone.0238673.g001

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