Bayesian Newton's Growth Model to examine infant's growth dynamics (trajectory, velocity, acceleration) and its association with prenatal environmental exposure.

1. Bayesian Analysis of Infant's Growth Dynamics with In
Utero Exposure to Environmental Toxicants
Jonggyu Baek∗, Bin Zhu, Peter X.K. Song
∗Postdoc Research Fellow: jongguri@umich.edu
Biostatistics, University of Michigan
7/30/2017
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2. Introduction
Endocrine disrupting compounds (EDCs)
EDC, such as phthalates, may aect lipid metabolism and adipocytes
(i.e., lipid system) and thus may also alter child's growth and
development.
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3. Introduction
EDCs growth development
Age at the infant BMI peak may be associated with later health
outcomes in childhood (e.g., obesity, tempo in sexual maturation).
Examining inter-relationship
between growth dynamics
and environmental exposure
is an important task in the
environmental health science.
Figure 1: BMI growth curves
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4. Introduction
Aim
Specic aim :
to develop a model for infant BMI growth dynamics (trajectory,
velocity, acceleration) in 0-3 years.
to investigate whether/how in utero exposures to BPA and phthalates
are associated with infant BMI growth acceleration.
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5. Data
ELEMENT data
The Early Life Exposure in Mexico to ENvironmental Toxicants (ELEMENT)
consists of three birth cohorts from Mexico City maternity hospitals.
The mother-child pairs of study participants were recruited over a series of years.
1994-1997
Cohort 1
1997-2000
Cohort 2
2001-2005
Cohort 5
C1-cohort1
From birth
until 48 mon
PL-plasma
From preg.
until 60 mon
BI-biomarker
From birth
until 60 mon
SF-SuperFund
From preg.
until 60 mon
Over 2000
subjects
250 preg. women for BMI
growth analysis
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6. Data
ELEMENT data
Outcome (Y): infant's BMI (kg/m2
) measured at-birth, 3, 6, 12, 18,
24, 30, 36 months.
Exposure (X): 10 single EDCs (BPA, 9 phthalates) and 3 mixtures of
EDCs, SumPCP (personal care product), SumDHEP(PVC plastics),
SumAA(anti-androgenic activity).
Confounders (Z): indicator of cohort, infant sex, breast-fed for 6
month, birth-weight, gestational age, mother's age at birth, marital
status, years of education, number of previous pregnancy,
current/previous smoking.
Excluded preterm ( 37 weeks), biologically implausible BMI Z-score
(|BMIz|5), all missing EDC during three trimesters.
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7. Data
Observed BMI in 0-3 years
Figure 2: Infant BMI trajectories over time (0, 3, 6, 12, 18, 24, 30, 36 months).
0.0 0.5 1.0 1.5 2.0 2.5 3.0
10121416182022
Age(year)
BMI
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8. Data
Challenges and solution
Observational times are relatively sparse (0, 3, 6, 12, 18, 24, 30, 36
months).
In intervals with no data recorded, WHO prior knowledge is applied to
constrain trajectory searching space and to smooth out potential
short-term uctuations.
A exible Bayesian model to capture growth dynamics (i.e., trajectory,
velocity, and acceleration) of BMI, and relate them to EDC toxicants.
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9. Method
Newton's Growth Model (NGM)
For each child i,



Yi (t) = Ui (t) + i (t), i (t) ∼ N(0, σ2
), (trajectory)
dUi (t) = Vi (t)dt, (velocity)
dVi (t) = a{Vi (t); Xi , θi }dt + b{Vi (t); Xi , θi }dWi (t), (acceleration)
where
i (t) ∼ N(0, σ2
),
a{Vi (t); Xi , θi }dt is the drift term for long term dynamics,
b{Vi (t); Xi , θi }dWi (t) is the diusion term for short term dynamics,
Wi (t) denotes the standard Wiener process dWi (t) ∼ N(0, dt).
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10. Method
Acceleration equation: Ornstein-Uhlenbeck (OU) process
Long term dynamics, a{Vi (t); Xi , θi } = −ρi (Vi (t) − ¯νi ), where
acceleration is characterized by rate ρi ( 0) and stable velocity ¯νi .
Short term dynamics, b{Vi (t); Xi , θi } = σξ is the variability of the
rate process.
Figure 3: Illustration of OU prior. A-B: σ2
ξ = 0, C-D: σ2
ξ = 5.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(year)
−1
4
9
14
19
V(t)
A
ρ = 7, ν = −0.5
ρ = 5, ν = −0.5
ρ = 7, ν = −1
ρ = 5, ν = −1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(year)
14
16
18
20
U(t)
B
ρ = 7, ν = −0.5
ρ = 5, ν = −0.5
ρ = 7, ν = −1
ρ = 5, ν = −1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(year)
−1
4
9
14
19
V(t)
C
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(year)
14
16
18
20
U(t)
D
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11. Method
Video clips: initial force for acceleration
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12. Method
Trajectory and velocity in infant BMI
Figure 4: Trajectory and velocity curve of BMI in the early life
Velocity
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13. Method
Prenatal exposure to EDCs into the initial force of
acceleration
For each child i with the jth exposure Xijt, j = 1, ..., 10 at trimester t, and
a set of confounders Zi , consider



log ρi = β0 + β1Xijt + Zi βz + 1i , 1i ∼ N(0, σ2
ρ).
¯νi = γ0 + Zi γz + 2i , 2i ∼ N(0, σ2
¯ν).
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14. Prior
WHO 2 month data
Figure 5: Fitted NGMs in the WHO 2-month boy/girl BMI data. Population
mean BMI trajectory for boys (A) and girls (C). Population mean BMI velocity
for boys (B) and girls (D).
q
q
q
q q
q
q
q
q
q
q
q q
0.0 0.5 1.0 1.5 2.0
131415161718
Age(year)
U(t)forBoys
A
q Mean BMI in WHO
0.0 0.5 1.0 1.5 2.0 2.5 3.0
010203040
Age(year)
V(t)forBoys
B
q
q
q
q q
q
q
q
q
q
q q q
0.0 0.5 1.0 1.5 2.0
131415161718
Age(year)
U(t)forGirls
C
q Mean BMI in WHO
0.0 0.5 1.0 1.5 2.0 2.5 3.0
010203040
Age(year)
V(t)forGirls D
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15. Prior
Priors
Prior for the initial distribution, (Ui (t0), Vi (t0)) ∼ N(m0, C0), plays
important roles in constraining trajectory searching space.
m0 = (13.2, 31) for boys and (13.1, 23) for girls.
Var[Ui (t0)] = 10 is 10 times greater than the WHO population
variance.
Var[Vi (t0)] = 22 is 2 times greater than the empirical variance (11).
Vague priors for σ2
, σ2
ρ, σ2
¯ν, σ2
ξ , [β0, β1, βz], [γ0, γz].
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16. Prior
NGM Analysis
Fitted models with dierent sets of Zi (sequentially adjusting cohort,
infant's sex, breast-fed, birth weight, gestational age, mother's age at
birth, marital status, education years, previous/current smoking).
Deviance Information criteria (DIC) indicated the best goodness-of-t
achieved at the model with adjusting cohort, infant's sex, breast fed
up to 6 months, birth weight, gestational age, mother's age at birth.
5000 MCMC iterations with the rst 1000 burin-in iterations.
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17. Data Analysis
NGM result
Figure 6:
(A) Estimated overall mean BMI trajectory (black bold line) and individual mean BMI
trajectories (grey dash lines).
(B) Estimated overall mean BMI velocity (black bold line) and acceleration (blue bold
line) with individual mean BMI velocities (grey dash lines).
(C) Histogram for timing of achieving infant BMI peak across infants.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(year)
10
15
20
25
U
^
(t)
q
q q q q q q
q
q Empirical mean BMI
Infant peak
(a)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(year)
0
10
20
30
V
^
(t)
−120
−80
−40
0
A
^
(t)
Acceleration = 0
(b)
Age(year)
Frequency
0.5 1.0 1.5 2.0 2.5 3.0
0
10
20
30
40
50
60 Median BMI peak = 0.9
Age at the BMI peak 3
(c)
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18. Data Analysis
NGM result: continued
The direction of eects of single/mixture of EDCs in 1st
trimester were all negative.
The inverse association results in delayed BMI peak.
SumAA in 1st
trimester and MiBP in 2nd
trimester were signicant.
Figure 7: Estimated ˆβ1 from NGMs
−0.2−0.10.00.10.2
Estimate
q q q q
q
q q q q
q
q
q q
qFirst trimester Second trimester Third trimester
BPA
M
BP
M
BzP
M
C
PP
M
EC
PP
M
EH
H
P
M
EH
P
M
EO
H
P
M
EP
M
iBP
Sum
PC
PSum
D
EH
PSum
AA
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19. Data Analysis
Comparison to SITAR
SuperImposition by Translation And Rotation (SITAR) model (Cole, 2010):
for infant i at time t,
E[Yit] = ai + h((t − bi )/exp(ci )),
where ai , bi , ci are infant-specic 'shift' and 'scale' random eects
corresponding to size, timing of peak velocity, and velocity,
respectively.
Model timing of peak, bi , to EDC Xijt: E[bi ] = δ0 + δ1Xijt + Zi δz.
Available R package, sitar, to t the above SITAR model.
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20. Data Analysis
SITAR result
Except BPA, MEP, and SumPCP, the estimated eects of single/mixture of EDCs
in the rst trimester were all positive.
The positive association results in delayed BMI infancy peak.
Figure 8: Estimated associations (ˆδ1) from the SITAR model with exposure to
EDCs for the rst trimester.
−0.2−0.10.00.10.20.30.4
Estimate
(b)
BPA
M
BP
M
BzP
M
C
PP
M
EC
PP
M
EH
H
P
M
EH
P
M
EO
H
P
M
EP
M
iBP
Sum
PC
PSum
D
EH
PSum
AA
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21. Data Analysis
Individual trajectories: NGM vs. SITAR
Figure 9: Estimated individual mean BMI trajectories from NGM and SITAR
model. Infant BMI peak was achieved at age around (a) 6 months, (b) 12
months, (c) 21 months, while (d) infant BMI steadily rose with no peak (or
delayed BMI peak after 3 years)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(Year)
q
q
q
q
q
q
q q
10
15
20
25
U
^
(t)
(a)
q
BMI trajectory (NGM)
BMI trajectory (SITAR)
Observed BMI
Infant peak
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(Year)
q
q
q
q
q
q
q q
10
15
20
25
U
^
(t)
(b)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(Year)
q
q
q
q
q
q
q
q
10
15
20
25
U
^
(t)
(c)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Age(Year)
q
q
q
q
q q
q
q
10
15
20
25
U
^
(t)
(d)
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22. Acknowledgment
Acknowledgment
Funding:
This research is supported by National Institutes of Health Grant
R01ES024732.
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23. End
Thank you
Question??
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