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Using the Children Weight Model
In bw: Dynamic Body Weight Models for Children and Adults
require("bw")
require("ggplot2")
In this vignette we explain how to use the model for children in R; we develop and explain the equations involved both for casual and advanced readers.
Contents
Usage in R {#user}
Inputs
The main inputs for the body weight change model in children are:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| age | Age (yrs) | No | - |
| sex | Either 'male' or 'female' | No | - |
As an example consider a 7 year old 'female':
female_model1 <- child_weight(age = 7, sex = "female")
Furthermore, the model allows the user to input Fat and Fat Free Mass composition of the body:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| FM | Fat Mass (kg) | Yes | Model estimate |
| FFM | Fat Free Mass (kg) | Yes | Model estimate |
For example, our female might have 19.9 kg of Fat Mass and 5.74 kg of Fat Free Mass:
female_model2 <- child_weight(age = 7, sex = "female", FM = 19.9, FFM = 5.74)
Energy intake can also be inputed as a vector of daily energy consumption:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| EI | Energy Intake per day | Yes | Model estimate |
female_model3 <- child_weight(age = 7, sex = "female", FM = 19.9, FFM = 5.74,
EI = seq(1600, 1750, length.out = 365))
Note that in the examples above, EIchange = seq(1600, 1750, length.out = 365) is inputed as a vector with each day representing the consumption reduction for that day. See Energy section for additional information.
Other (optional) inputs include:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| days | Number of days to run de model | Yes | $365$ |
| dt | Time step for Rungue-Kutta 4 | Yes | $1$ |
| checkValues | Boolean indicating whether the model should check parameters make sense | Yes | TRUE |
All inputs used in the model are:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| age | Age (yrs) | No | - |
| sex | Either 'male' or 'female' | No | - |
| FM | Fat Mass (kg) | Yes | Model estimate |
| FFM | Fat Free Mass (kg) | Yes | Model estimate |
| EI | Energy Intake per day | Yes | Model estimate |
| days | Time period (days) to run the model | Yes | $365$ |
| dt | Time step for Rungue-Kutta 4 | Yes | $1$ |
| checkValues | Check for internal consistency | Yes | TRUE |
Finally, we remark that one can also input data from a database to estimate individual-level weight change (see the related section)
#Database information
ages <- c(8, 10, 7, 7, 12)
sexes <- c("male", "female", "female", "male", "male")
#Returns a weight change matrix and other matrices
database_model <- child_weight(ages, sexes)
Plots
Result plots can be obtained by model_plot function:
model_plot(female_model2, "Body_Weight")
Plotting options include "Body_Weight", Fat_Mass, and Fat_Free_Mass. Several can be chosen at the same time:
model_plot(female_model2, c("Body_Weight", "Fat_Mass"))
Variables can also be plotted against age:
model_plot(female_model2, c("Body_Weight", "Fat_Mass"), timevar = "Age")
Energy
Energy intake is usually not continuously measured but measured at different and distant points in time (say 1 year apart). The function energy_build allows the user to interpolate different energy models between the interpolation points. As an example consider an individual that initially consumed 1600 kcals, by day 365 consumed in 1750 kcals and by day 730 had increased his consumption to 1820 kcals. The energy_build function interpolates those values via a Brownian Bridge:
EIbrownian <- energy_build(c(1600, 1750, 1820), c(0, 365, 730))
The interpolation looks like this:
ggplot() + geom_line(aes(x = 1:730, y = EI), data = data.frame(EI = EIbrownian)) +
theme_classic() +
xlab("Days") + ylab("Energy intake (kcals)") + ggtitle("Energy interpolation")
Such energy change matrix can be directly inputed in the model:
model_brownian <- child_weight(10, "male", EI = EIbrownian, days = 730)
Other interpolation modes include Linear, Exponential, Stepwise_R (right stepwise), Stepwise_L (left stepwise), and Logarithmic:
EIlinear <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Linear")
EIexponential <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Exponential")
EIstepwise_r <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Stepwise_R")
EIstepwise_l <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Stepwise_L")
EIlogarithmic <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Logarithmic")
Which look like this:
ggplot() +
geom_line(aes(x = 1:730, y = EI, color = "Brownian"),
data = data.frame(EI = EIbrownian)) +
geom_line(aes(x = 1:730, y = EI, color = "Linear"),
data = data.frame(EI = EIlinear)) +
geom_line(aes(x = 1:730, y = EI, color = "Exponential"),
data = data.frame(EI = EIexponential)) +
geom_step(aes(x = 1:730, y = EI, color = "Right Stepwise"),
data = data.frame(EI = EIstepwise_r)) +
geom_step(aes(x = 1:730, y = EI, color = "Left Stepwise"),
data = data.frame(EI = EIstepwise_l)) +
geom_line(aes(x = 1:730, y = EI, color = "Logarithmic"),
data = data.frame(EI = EIlogarithmic)) +
xlab("Days") + ylab("Energy change (kcals)") +
ggtitle("Energy interpolation") +
theme_classic() +
scale_color_manual("Interpolation",
values = c("Brownian" = "red", "Linear" = "deepskyblue3",
"Exponential" = "forestgreen", "Logarithmic" = "purple",
"Right Stepwise" = "black", "Left Stepwise" = "green"))
These models result in different weight changes:
model_linear <- child_weight(10, "male", EI = EIlinear, days = 730)
model_exponential <- child_weight(10, "male", EI = EIexponential, days = 730)
model_logarithmic <- child_weight(10, "male", EI = EIlogarithmic, days = 730)
model_stepwise_r <- child_weight(10, "male", EI = EIstepwise_r, days = 730)
model_stepwise_l <- child_weight(10, "male", EI = EIstepwise_l, days = 730)
Which look as follows:
ggplot() +
geom_line(aes(x = 1:730, y = as.vector(model_linear[["Body_Weight"]]), color = "Linear")) +
geom_line(aes(x = 1:730, y = as.vector(model_exponential[["Body_Weight"]]), color = "Exponential")) +
geom_line(aes(x = 1:730, y = as.vector(model_logarithmic[["Body_Weight"]]), color = "Logarithmic")) +
geom_line(aes(x = 1:730, y = as.vector(model_stepwise_r[["Body_Weight"]]), color = "Right Stepwise")) +
geom_line(aes(x = 1:730, y = as.vector(model_stepwise_l[["Body_Weight"]]), color = "Left Stepwise")) +
geom_line(aes(x = 1:730, y = as.vector(model_brownian[["Body_Weight"]]), color = "Brownian")) +
xlab("Days") + ylab("Weight (kg)") +
theme_classic()+
ggtitle("Weight change under different energy interpolations") +
scale_color_manual("Interpolation",
values = c("Brownian" = "red", "Linear" = "deepskyblue3",
"Exponential" = "forestgreen", "Logarithmic" = "purple",
"Right Stepwise" = "black", "Left Stepwise" = "green"))
Using Richardson's function
The children model includes the option richardsonparams which is a list of parameters $list(K = NA, Q = NA, B = NA, A = NA, nu = NA, C = NA)$ representing $K, Q, \beta, A, \nu, C$. If parameters are specified, the model assumes the energy intake function is a generalized logistic function (Richardson's function @falkner2012human):
\begin{equation}
EI(t) = A + \frac{K-A}{(C + Q e^{-\beta \cdot t})^{1/\nu}}.
\end{equation}
girl <- child_weight(6,"female", days=365, dt = 5,
richardsonparams = list(K = 2700, Q = 10,
B = 12, A = 3, nu = 4,
C = 1))
model_plot(girl, "Body_Weight")
Estimating weight change of a database
Vector data can also be used in the model to calculate weight change for several individuals at a time (which is quite faster than doing them individually). As an example consider the following dataset:
#Database information
mydata <- data.frame(
id = 1:5,
age = c(8, 10, 7, 7, 12),
sex = c("male", "female", "female", "male", "male"),
energy = runif(5, 1500, 2000),
prob = c(0.1, 0.2, 0.2, 0.05, 0.45))
#Get energy change with energy build function
eichange <- energy_build(cbind(runif(5, 1500, 2000), mydata$energy), c(0, 365))
#Returns a weight change matrix and other matrices
database_model <- child_weight(mydata$age, mydata$sex, EI = t(eichange))
Plots can also be obtained for the population with the same command model_plot:
model_plot(database_model, "Body_Weight")
Summary measures can be obtained via model_mean which quantifies mean for 'Body_Weight', 'Fat_Free_Mass', and 'Fat_Mass':
model_mean(database_model, "Body_Weight")
head(model_mean(database_model, "Body_Weight"))[,1:5]
Mean is only estimated for some points in time, to estimate mean for the whole period, consider changing the days vector variable:
model_mean(database_model, "Body_Weight", days = 1:365)
head(model_mean(database_model, "Body_Weight", days = 1:365))[,1:5]
Mean can also be grouped by a variable (say, sex):
model_mean(database_model, "Body_Weight", days = 1:365, group = mydata$sex)
head(model_mean(database_model, "Body_Weight", days = 1:365, group = mydata$sex))[,1:5]
Finally, model_mean can also be used to estimate survey means using the svydesign from the survey package:
require("survey")
design <- svydesign(ids = ~id, probs = ~prob, data = mydata)
model_mean(database_model, group = mydata$sex, design = design)
require("survey")
design <- svydesign(ids = ~id, probs = ~prob, data = mydata)
head(model_mean(database_model, group = mydata$sex, design = design))[,1:5]
Additional information on the model for adults and other package functions can be obtained in the other package's Vignettes
browseVignettes("bw")
Basic explanation on model's equations {#basic}
The main idea of the children's weight model [@katan2016impact, @hall2013dynamics] is that weight is determined by fat and fat free mass:
\begin{equation}
\underbrace{BW(t)}{\text{Body Weight}} = \underbrace{FM(t)}{\text{Fat Mass}} + \underbrace{FFM(t)}{\text{Fat Free Mass}}.
\end{equation}
Both fat and fat free masses are interdependent; their dependencies given by the equation system:
\begin{equation}
\begin{aligned}
\overbrace{\dfrac{d FFM}{dt}}^{\text{Change in fat free mass}} & = \overbrace{\dfrac{1}{\rho{FFM}}}^{\text{Scaling}} \times \overbrace{p}^{\text{Proportion of energy difference going to fat free}} \times \overbrace{(EI - E)}^{\text{Difference between intake and expenditure}}
\ & \qquad + \underbrace{g}{\text{Control for growth}} \\
\underbrace{\dfrac{d FM}{dt}}{\text{Change in fat mass}} & = \underbrace{\dfrac{1}{\rho_{FM}}}{\text{Scaling}} \times \underbrace{(1-p)}{\text{Proportion of energy difference going to fat}} \times \underbrace{(EI - E)}{\text{Difference between intake and expenditure}}
\ & \qquad - \underbrace{g}{\text{Control for growth}}
\end{aligned}
\end{equation}
Energy intake $EI$ is a function that can either be specified by the user (via the EI entry) or one can use a Richardson's function
\begin{equation}
EI(t) = A + \frac{K-A}{(C + Q e^{-\beta \cdot t})^{1/\nu}}
\end{equation}
by specifying the individual's parameters $K,A,C,Q,\beta,\nu$. The energy expenditure is given by a combination of the changes in fat and fat free masses as well as the masses themselves:
\begin{equation}
\begin{aligned}
E & = \overbrace{K}{\text{Individual's parameter}} + \overbrace{\gamma{FFM} FFM}^{\text{Fat Free Mass}} + \overbrace{\gamma_{FM} FM}^{\text{Fat Mass}} + \overbrace{\beta \Delta I}^{\text{Difference in intake with respect to normal growth}} \\\ & \qquad + \underbrace{\delta \cdot BW}{\text{Body weight}} + \underbrace{\eta{FFM} \cdot \dfrac{dFFM}{dt} + \eta_{FM} \cdot \dfrac{dFM}{dt}}_{\text{Changes in fat free and fat masses}},
\end{aligned}
\end{equation}
Thus body weight is the result of this physiological interdependencies between the fat mass, the fat free mass, the energy intake and the expenditure. The model is further expanded in the following section.
Advanced explanation on model's equations {#advanced}
The children weight model [@katan2016impact, @hall2013dynamics] considers the interactions between fat mass $FM := FM(t)$; fat free mass $FFM := FFM(t)$; an energy intake function $EI:= EI(t)$; an energy expenditure function $E := E(t)$ adjusted by a body-growth term $g(t)$. In this model, body weight is given by the sum of fat mass and fat free mass:
\begin{equation}
BW:= BW(t) = FM(t) + FFM(t).
\end{equation}
In particular, the body weight function (as a function of time $t$) depends on the individual's sex ($\textrm{Sex}$), initial fat mass ($FM_0$), initial fat free mass ($FFM_0$), energy consumption function ($EI(t)$).
The components of body weight, $FM$ and $FFM$, are determined by a system of ordinary differential equations:
\begin{equation}\label{diff}
\begin{aligned}
\rho_{FFM} \cdot \dfrac{d FFM}{dt} & = p \cdot (EI - E) + g \\
\rho_{FM} \cdot \dfrac{d FM}{dt} & = (1-p) \cdot (EI - E) - g
\end{aligned}
\end{equation}
where $p = C/(C + FM)$ a ratio established by Forbes with $C = 10.4 \rho_{FFM} / \rho_{FM}$. The parameters $\rho_{FM} = 9.4 kcal/g$ ($= 9400 \textrm{ kcal}/\textrm{kg}$ and $\rho_{FFM} = (4.3 \cdot FFM_0 + 837) \textrm{ kcal}/\textrm{kg}$ where $FFM_0$ represents the amount of fat free mass (kg) at baseline.
The growth term is given by:
\begin{equation}\label{growth}
g:= g(t) = A e^{-(t - t_{A})/\tau_{A}} + B e^{-(t - t_{B})^2/2\tau_{B}^2} + D e^{-(t - t_{D})^2/2\tau_{D}^2},
\end{equation}
where the parameters for each sex are given by table \ref{gparams}.
\begin{table}[!htb]
\centering
\begin{tabular}{lrrr}
\bf Parameter & \bf Males & \bf Females & \bf Scale \
\hline
\hline
$A$ & 3.2 & 2.3 & kcal/day \
$B$ & 9.6 & 8.4 & kcal/day\
$D$ & 10.1 & 1.1 & kcal/day\
$\tau_{A}$ & 2.5 & 1 & years\
$\tau_{B}$ & 9.6 & 8.4 & years \
$\tau_{D}$ & 1.5 & 0.7 & years \
$t_{A}$ & 4.7 & 4.5 & years \
$t_{B}$ & 12.5 & 11.7 & years \
$t_{D}$ & 15 & 16.2 & years \
\hline
\hline
\end{tabular}
\caption{Parameters for the growth function $g$ as established in \eqref{growth}}
\label{gparams}
\end{table}
Energy expenditure in \eqref{energy} is given by:
\begin{equation}\label{energy}
E = K + \gamma_{FFM} FFM + \gamma_{FM} FM + \beta \Delta I + \delta \cdot BW + \eta_{FFM} \cdot \dfrac{dFFM}{dt} + \eta_{FM} \cdot \dfrac{dFM}{dt},
\end{equation}
where $K$ represents an energy expenditure constant determined by the initial energy balance ($K = 800 \textrm{ kcal}/\textrm{d}$ for males; $K = 700 \textrm{ kcal}/\textrm{d}$ for females); $\beta = 0.24$ stands for the adaptation of energy expenditure when intake is perturbed $\Delta I$; $\eta_{FM} = 180 \textrm{ kcal}/\textrm{kg}$ and $\eta_{FFM} = 230 \textrm{ kcal}/\textrm{kg}$ account for ``biochemical efficiencies associated to fat and protein synthesis'' \cite{hall2013dynamics}.
In particular, the function for physical activity $\delta$ in \eqref{energy} is given by:
\begin{equation}
\delta(t) = \delta_{min} + \dfrac{(\delta_{max}-\delta_{min}) P^{h}}{t^h + P^h}
\end{equation}
with $\delta_{min} = 10 \textrm{ kcal}/\textrm{kg}/\textrm{d}$ represents the average activity of young adults whilst $\delta_{max}$ ($\delta_{max} = 19 \textrm{ kcal}/\textrm{kg}/\textrm{d}$ and $\delta_{max} = 17 \textrm{ kcal}/\textrm{kg}/\textrm{d}$ for males and females respectively) stands for the average physical activity for young children. The parameter $P = 12 \textrm{ years}$ stands for the point of maximum physical activity whilst the constant $h = 10$ represents the rate of decline as a function of age.
The perturbation of energy intake $\Delta I$ represents the shift away from the intake associated to normal growth ($\Delta I$ in \eqref{energy}). It was estimated by:
\begin{equation}
\Delta I(t) = EI(t) - I_{ref}(t)
\end{equation}
where:
\begin{equation}\label{iref}
\begin{aligned}
I_{ref}(t) & = EB_{ref} + K + (\gamma_{FFM} + \delta) FFM_{ref} + (\gamma_{FM} + \delta) FM_{ref} + \dfrac{\eta_{FFM}}{\rho_{FFM}} (p\cdot EB_{ref} + g) \ & \qquad + \dfrac{\eta_{FM}}{\rho_{FM}} \big((1-p)\cdot EB_{ref} + g \big)
\end{aligned}
\end{equation}
represents the reference energy intake for normal growth. The energy balance for reference children is also of the form:
\begin{equation}
EB_{ref}(t) = A e^{-(t - t^{EB}{A})/\tau^{EB}{A}} + B e^{-(t - t^{EB}{B})^2/2(\tau^{EB}{B})^2} + D e^{-(t - t^{EB}{D})^2/2(\tau{D}^{EB})^2}.
\end{equation}
The reference fat free mass $FFM_{ref}$ and the reference fat mass $FM_{ref}$ in \eqref{iref} were obtained from linear interpolations to the values in table \ref{ref} which were obtained from [@ellis2000reference, @fomon1982body, @haschke1989body].
\begin{table}[!htb]
\centering
\begin{tabular}{lrrcrr}
& \multicolumn{2}{c}{\bf Males} & & \multicolumn{2}{c}{\bf Females} \
\cmidrule{2-3}
\cmidrule{5-6}
Age & Fat Mass (kg) & Fat Free Mass (kg) & & Fat Mass (kg) & Fat Free Mass (kg)\
\hline
\hline
4 & 14.0 & 2.7 & & 13.2 & 2.8 \
5 & 16.0 & 2.7 & & 14.7 & 2.9 \
6 & 17.9 & 2.8 & & 16.3 & 3.2 \
7 & 19.9 & 2.9 & & 18.2 & 3.7 \
8 & 22.0 & 3.3 & & 20.5 & 4.3 \
9 & 24.4 & 3.7 & & 23.3 & 5.2 \
10 & 27.5 & 4.8 & & 26.4 & 7.2 \
11 & 29.5 & 5.9 & & 28.5 & 8.5 \
12 & 33.2 & 6.7 & & 32.4 & 9.2 \
13 & 38.1 & 7.0 & & 36.1 & 10.0 \
14 & 43.6 & 7.2 & & 38.9 & 11.3 \
15 & 49.1 & 7.5 & & 40.7 & 12.8 \
16 & 54.0 & 8.0 & & 41.7 & 14.0 \
17 & 57.7 & 8.4 & & 42.3 & 14.3 \
18 & 60.0 & 8.8 & & 42.6 & 14.3 \
\hline
\hline
\end{tabular}
\caption{Reference values for fat and fat free mass (kg) from [@ellis2000reference, @fomon1982body, @haschke1989body] for interpolation to obtain $FM_{ref}$ and $FFM_{ref}$.}
\label{ref}
\end{table}
A closed form expression for the energy is given by:
\begin{equation}
E = \dfrac{K + (\gamma_{FFM} + \delta) FFM + (\gamma_{FM} + \delta) FM + \beta \cdot \Delta I + \Big( \frac{\eta_{FFM}}{\rho_{FFM}} p + \frac{\eta_{FM}}{\rho_{FM}}\cdot (1-p) \Big) \cdot EI + g \cdot \Big(\frac{\eta_{FFM}}{\rho_{FFM}} - \frac{\eta_{FM}}{\rho_{FM}} \Big)}{1 + \frac{\eta_{FFM}}{\rho_{FFM}} p + \frac{\eta_{FM}}{\rho_{FM}}\cdot (1-p) }
\end{equation}
\subsubsection{Initial values of system of ordinary differential equations}
To obtain the initial values for \eqref{diff} we estimated the initial fat mass using the equations from @deurenberg1991body:
\begin{equation}\label{fm}
FM_{0} = \begin{cases}
1.51 \cdot \text{BMI}_0 - 0.7 \cdot a - \frac{2.2}{100} \cdot BW_0 & \textrm{ if Male}\\
1.51 \cdot BMI_0 - 0.7 \cdot a + \frac{1.4}{100} \cdot BW_0 & \textrm{ if Female}
\end{cases}
\end{equation}
where $a$ represents the age (years); $\text{BMI}$ the initial body mass index ($\text{kg}/\text{m}^2$) and $BW_0$ the initial body weight of the child. The initial fat free mass is given by the difference between fat mass and body weight:
\begin{equation}\label{ffm}
FFM_0 = BW_0 - FM_0.
\end{equation}
Solution algorithm
To solve this system of differential equations, we used a 4th order Runge-Kutta algorithm (RK4) @ascher2011first with a stepsize $\Delta t = 1$. RK4 was programmed in C++ for speed and connected to R via the Rcpp package [@Rcpp2, @Rcpp1].
Additional information
Additional information on the adult's model, and why to use the dynamic adult model instead of other classical approaches can be found in the package's vignettes:
browseVignettes("bw")
References
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require("bw") require("ggplot2")
In this vignette we explain how to use the model for children in R; we develop and explain the equations involved both for casual and advanced readers.
Contents
Usage in R {#user}
Inputs
The main inputs for the body weight change model in children are:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| age | Age (yrs) | No | - |
| sex | Either 'male' or 'female' | No | - |
As an example consider a 7 year old 'female':
female_model1 <- child_weight(age = 7, sex = "female")
Furthermore, the model allows the user to input Fat and Fat Free Mass composition of the body:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| FM | Fat Mass (kg) | Yes | Model estimate |
| FFM | Fat Free Mass (kg) | Yes | Model estimate |
For example, our female might have 19.9 kg of Fat Mass and 5.74 kg of Fat Free Mass:
female_model2 <- child_weight(age = 7, sex = "female", FM = 19.9, FFM = 5.74)
Energy intake can also be inputed as a vector of daily energy consumption:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| EI | Energy Intake per day | Yes | Model estimate |
female_model3 <- child_weight(age = 7, sex = "female", FM = 19.9, FFM = 5.74, EI = seq(1600, 1750, length.out = 365))
Note that in the examples above, EIchange = seq(1600, 1750, length.out = 365) is inputed as a vector with each day representing the consumption reduction for that day. See Energy section for additional information.
Other (optional) inputs include:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| days | Number of days to run de model | Yes | $365$ |
| dt | Time step for Rungue-Kutta 4 | Yes | $1$ |
| checkValues | Boolean indicating whether the model should check parameters make sense | Yes | TRUE |
All inputs used in the model are:
| Input | Meaning | Optional | Default |
|------:|------:|------:|------:|
| age | Age (yrs) | No | - |
| sex | Either 'male' or 'female' | No | - |
| FM | Fat Mass (kg) | Yes | Model estimate |
| FFM | Fat Free Mass (kg) | Yes | Model estimate |
| EI | Energy Intake per day | Yes | Model estimate |
| days | Time period (days) to run the model | Yes | $365$ |
| dt | Time step for Rungue-Kutta 4 | Yes | $1$ |
| checkValues | Check for internal consistency | Yes | TRUE |
Finally, we remark that one can also input data from a database to estimate individual-level weight change (see the related section)
#Database information ages <- c(8, 10, 7, 7, 12) sexes <- c("male", "female", "female", "male", "male") #Returns a weight change matrix and other matrices database_model <- child_weight(ages, sexes)
Plots
Result plots can be obtained by model_plot function:
model_plot(female_model2, "Body_Weight")
Plotting options include "Body_Weight", Fat_Mass, and Fat_Free_Mass. Several can be chosen at the same time:
model_plot(female_model2, c("Body_Weight", "Fat_Mass"))
Variables can also be plotted against age:
model_plot(female_model2, c("Body_Weight", "Fat_Mass"), timevar = "Age")
Energy
Energy intake is usually not continuously measured but measured at different and distant points in time (say 1 year apart). The function energy_build allows the user to interpolate different energy models between the interpolation points. As an example consider an individual that initially consumed 1600 kcals, by day 365 consumed in 1750 kcals and by day 730 had increased his consumption to 1820 kcals. The energy_build function interpolates those values via a Brownian Bridge:
EIbrownian <- energy_build(c(1600, 1750, 1820), c(0, 365, 730))
The interpolation looks like this:
ggplot() + geom_line(aes(x = 1:730, y = EI), data = data.frame(EI = EIbrownian)) + theme_classic() + xlab("Days") + ylab("Energy intake (kcals)") + ggtitle("Energy interpolation")
Such energy change matrix can be directly inputed in the model:
model_brownian <- child_weight(10, "male", EI = EIbrownian, days = 730)
Other interpolation modes include Linear, Exponential, Stepwise_R (right stepwise), Stepwise_L (left stepwise), and Logarithmic:
EIlinear <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Linear") EIexponential <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Exponential") EIstepwise_r <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Stepwise_R") EIstepwise_l <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Stepwise_L") EIlogarithmic <- energy_build(c(1600, 1750, 1820), c(0, 365, 730), "Logarithmic")
Which look like this:
ggplot() + geom_line(aes(x = 1:730, y = EI, color = "Brownian"), data = data.frame(EI = EIbrownian)) + geom_line(aes(x = 1:730, y = EI, color = "Linear"), data = data.frame(EI = EIlinear)) + geom_line(aes(x = 1:730, y = EI, color = "Exponential"), data = data.frame(EI = EIexponential)) + geom_step(aes(x = 1:730, y = EI, color = "Right Stepwise"), data = data.frame(EI = EIstepwise_r)) + geom_step(aes(x = 1:730, y = EI, color = "Left Stepwise"), data = data.frame(EI = EIstepwise_l)) + geom_line(aes(x = 1:730, y = EI, color = "Logarithmic"), data = data.frame(EI = EIlogarithmic)) + xlab("Days") + ylab("Energy change (kcals)") + ggtitle("Energy interpolation") + theme_classic() + scale_color_manual("Interpolation", values = c("Brownian" = "red", "Linear" = "deepskyblue3", "Exponential" = "forestgreen", "Logarithmic" = "purple", "Right Stepwise" = "black", "Left Stepwise" = "green"))
These models result in different weight changes:
model_linear <- child_weight(10, "male", EI = EIlinear, days = 730) model_exponential <- child_weight(10, "male", EI = EIexponential, days = 730) model_logarithmic <- child_weight(10, "male", EI = EIlogarithmic, days = 730) model_stepwise_r <- child_weight(10, "male", EI = EIstepwise_r, days = 730) model_stepwise_l <- child_weight(10, "male", EI = EIstepwise_l, days = 730)
Which look as follows:
ggplot() + geom_line(aes(x = 1:730, y = as.vector(model_linear[["Body_Weight"]]), color = "Linear")) + geom_line(aes(x = 1:730, y = as.vector(model_exponential[["Body_Weight"]]), color = "Exponential")) + geom_line(aes(x = 1:730, y = as.vector(model_logarithmic[["Body_Weight"]]), color = "Logarithmic")) + geom_line(aes(x = 1:730, y = as.vector(model_stepwise_r[["Body_Weight"]]), color = "Right Stepwise")) + geom_line(aes(x = 1:730, y = as.vector(model_stepwise_l[["Body_Weight"]]), color = "Left Stepwise")) + geom_line(aes(x = 1:730, y = as.vector(model_brownian[["Body_Weight"]]), color = "Brownian")) + xlab("Days") + ylab("Weight (kg)") + theme_classic()+ ggtitle("Weight change under different energy interpolations") + scale_color_manual("Interpolation", values = c("Brownian" = "red", "Linear" = "deepskyblue3", "Exponential" = "forestgreen", "Logarithmic" = "purple", "Right Stepwise" = "black", "Left Stepwise" = "green"))
Using Richardson's function
The children model includes the option richardsonparams which is a list of parameters $list(K = NA, Q = NA, B = NA, A = NA, nu = NA, C = NA)$ representing $K, Q, \beta, A, \nu, C$. If parameters are specified, the model assumes the energy intake function is a generalized logistic function (Richardson's function @falkner2012human):
\begin{equation}
EI(t) = A + \frac{K-A}{(C + Q e^{-\beta \cdot t})^{1/\nu}}.
\end{equation}
girl <- child_weight(6,"female", days=365, dt = 5, richardsonparams = list(K = 2700, Q = 10, B = 12, A = 3, nu = 4, C = 1)) model_plot(girl, "Body_Weight")
Estimating weight change of a database
Vector data can also be used in the model to calculate weight change for several individuals at a time (which is quite faster than doing them individually). As an example consider the following dataset:
#Database information mydata <- data.frame( id = 1:5, age = c(8, 10, 7, 7, 12), sex = c("male", "female", "female", "male", "male"), energy = runif(5, 1500, 2000), prob = c(0.1, 0.2, 0.2, 0.05, 0.45)) #Get energy change with energy build function eichange <- energy_build(cbind(runif(5, 1500, 2000), mydata$energy), c(0, 365)) #Returns a weight change matrix and other matrices database_model <- child_weight(mydata$age, mydata$sex, EI = t(eichange))
Plots can also be obtained for the population with the same command model_plot:
model_plot(database_model, "Body_Weight")
Summary measures can be obtained via model_mean which quantifies mean for 'Body_Weight', 'Fat_Free_Mass', and 'Fat_Mass':
model_mean(database_model, "Body_Weight")
head(model_mean(database_model, "Body_Weight"))[,1:5]
Mean is only estimated for some points in time, to estimate mean for the whole period, consider changing the days vector variable:
model_mean(database_model, "Body_Weight", days = 1:365)
head(model_mean(database_model, "Body_Weight", days = 1:365))[,1:5]
Mean can also be grouped by a variable (say, sex):
model_mean(database_model, "Body_Weight", days = 1:365, group = mydata$sex)
head(model_mean(database_model, "Body_Weight", days = 1:365, group = mydata$sex))[,1:5]
Finally, model_mean can also be used to estimate survey means using the svydesign from the survey package:
require("survey") design <- svydesign(ids = ~id, probs = ~prob, data = mydata) model_mean(database_model, group = mydata$sex, design = design)
require("survey") design <- svydesign(ids = ~id, probs = ~prob, data = mydata) head(model_mean(database_model, group = mydata$sex, design = design))[,1:5]
Additional information on the model for adults and other package functions can be obtained in the other package's Vignettes
browseVignettes("bw")
Basic explanation on model's equations {#basic}
The main idea of the children's weight model [@katan2016impact, @hall2013dynamics] is that weight is determined by fat and fat free mass:
\begin{equation}
\underbrace{BW(t)}{\text{Body Weight}} = \underbrace{FM(t)}{\text{Fat Mass}} + \underbrace{FFM(t)}{\text{Fat Free Mass}}.
\end{equation}
Both fat and fat free masses are interdependent; their dependencies given by the equation system:
\begin{equation}
\begin{aligned}
\overbrace{\dfrac{d FFM}{dt}}^{\text{Change in fat free mass}} & = \overbrace{\dfrac{1}{\rho{FFM}}}^{\text{Scaling}} \times \overbrace{p}^{\text{Proportion of energy difference going to fat free}} \times \overbrace{(EI - E)}^{\text{Difference between intake and expenditure}}
\ & \qquad + \underbrace{g}{\text{Control for growth}} \\
\underbrace{\dfrac{d FM}{dt}}{\text{Change in fat mass}} & = \underbrace{\dfrac{1}{\rho_{FM}}}{\text{Scaling}} \times \underbrace{(1-p)}{\text{Proportion of energy difference going to fat}} \times \underbrace{(EI - E)}{\text{Difference between intake and expenditure}}
\ & \qquad - \underbrace{g}{\text{Control for growth}}
\end{aligned}
\end{equation}
Energy intake $EI$ is a function that can either be specified by the user (via the EI entry) or one can use a Richardson's function
\begin{equation}
EI(t) = A + \frac{K-A}{(C + Q e^{-\beta \cdot t})^{1/\nu}}
\end{equation}
by specifying the individual's parameters $K,A,C,Q,\beta,\nu$. The energy expenditure is given by a combination of the changes in fat and fat free masses as well as the masses themselves:
\begin{equation}
\begin{aligned}
E & = \overbrace{K}{\text{Individual's parameter}} + \overbrace{\gamma{FFM} FFM}^{\text{Fat Free Mass}} + \overbrace{\gamma_{FM} FM}^{\text{Fat Mass}} + \overbrace{\beta \Delta I}^{\text{Difference in intake with respect to normal growth}} \\\ & \qquad + \underbrace{\delta \cdot BW}{\text{Body weight}} + \underbrace{\eta{FFM} \cdot \dfrac{dFFM}{dt} + \eta_{FM} \cdot \dfrac{dFM}{dt}}_{\text{Changes in fat free and fat masses}},
\end{aligned}
\end{equation}
Thus body weight is the result of this physiological interdependencies between the fat mass, the fat free mass, the energy intake and the expenditure. The model is further expanded in the following section.
Advanced explanation on model's equations {#advanced}
The children weight model [@katan2016impact, @hall2013dynamics] considers the interactions between fat mass $FM := FM(t)$; fat free mass $FFM := FFM(t)$; an energy intake function $EI:= EI(t)$; an energy expenditure function $E := E(t)$ adjusted by a body-growth term $g(t)$. In this model, body weight is given by the sum of fat mass and fat free mass: \begin{equation} BW:= BW(t) = FM(t) + FFM(t). \end{equation} In particular, the body weight function (as a function of time $t$) depends on the individual's sex ($\textrm{Sex}$), initial fat mass ($FM_0$), initial fat free mass ($FFM_0$), energy consumption function ($EI(t)$).
The components of body weight, $FM$ and $FFM$, are determined by a system of ordinary differential equations: \begin{equation}\label{diff} \begin{aligned} \rho_{FFM} \cdot \dfrac{d FFM}{dt} & = p \cdot (EI - E) + g \\ \rho_{FM} \cdot \dfrac{d FM}{dt} & = (1-p) \cdot (EI - E) - g \end{aligned} \end{equation} where $p = C/(C + FM)$ a ratio established by Forbes with $C = 10.4 \rho_{FFM} / \rho_{FM}$. The parameters $\rho_{FM} = 9.4 kcal/g$ ($= 9400 \textrm{ kcal}/\textrm{kg}$ and $\rho_{FFM} = (4.3 \cdot FFM_0 + 837) \textrm{ kcal}/\textrm{kg}$ where $FFM_0$ represents the amount of fat free mass (kg) at baseline.
The growth term is given by: \begin{equation}\label{growth} g:= g(t) = A e^{-(t - t_{A})/\tau_{A}} + B e^{-(t - t_{B})^2/2\tau_{B}^2} + D e^{-(t - t_{D})^2/2\tau_{D}^2}, \end{equation} where the parameters for each sex are given by table \ref{gparams}.
\begin{table}[!htb] \centering \begin{tabular}{lrrr} \bf Parameter & \bf Males & \bf Females & \bf Scale \ \hline \hline $A$ & 3.2 & 2.3 & kcal/day \ $B$ & 9.6 & 8.4 & kcal/day\ $D$ & 10.1 & 1.1 & kcal/day\ $\tau_{A}$ & 2.5 & 1 & years\ $\tau_{B}$ & 9.6 & 8.4 & years \ $\tau_{D}$ & 1.5 & 0.7 & years \ $t_{A}$ & 4.7 & 4.5 & years \ $t_{B}$ & 12.5 & 11.7 & years \ $t_{D}$ & 15 & 16.2 & years \ \hline \hline \end{tabular} \caption{Parameters for the growth function $g$ as established in \eqref{growth}} \label{gparams} \end{table}
Energy expenditure in \eqref{energy} is given by: \begin{equation}\label{energy} E = K + \gamma_{FFM} FFM + \gamma_{FM} FM + \beta \Delta I + \delta \cdot BW + \eta_{FFM} \cdot \dfrac{dFFM}{dt} + \eta_{FM} \cdot \dfrac{dFM}{dt}, \end{equation} where $K$ represents an energy expenditure constant determined by the initial energy balance ($K = 800 \textrm{ kcal}/\textrm{d}$ for males; $K = 700 \textrm{ kcal}/\textrm{d}$ for females); $\beta = 0.24$ stands for the adaptation of energy expenditure when intake is perturbed $\Delta I$; $\eta_{FM} = 180 \textrm{ kcal}/\textrm{kg}$ and $\eta_{FFM} = 230 \textrm{ kcal}/\textrm{kg}$ account for ``biochemical efficiencies associated to fat and protein synthesis'' \cite{hall2013dynamics}.
In particular, the function for physical activity $\delta$ in \eqref{energy} is given by: \begin{equation} \delta(t) = \delta_{min} + \dfrac{(\delta_{max}-\delta_{min}) P^{h}}{t^h + P^h} \end{equation} with $\delta_{min} = 10 \textrm{ kcal}/\textrm{kg}/\textrm{d}$ represents the average activity of young adults whilst $\delta_{max}$ ($\delta_{max} = 19 \textrm{ kcal}/\textrm{kg}/\textrm{d}$ and $\delta_{max} = 17 \textrm{ kcal}/\textrm{kg}/\textrm{d}$ for males and females respectively) stands for the average physical activity for young children. The parameter $P = 12 \textrm{ years}$ stands for the point of maximum physical activity whilst the constant $h = 10$ represents the rate of decline as a function of age.
The perturbation of energy intake $\Delta I$ represents the shift away from the intake associated to normal growth ($\Delta I$ in \eqref{energy}). It was estimated by:
\begin{equation}
\Delta I(t) = EI(t) - I_{ref}(t)
\end{equation}
where:
\begin{equation}\label{iref}
\begin{aligned}
I_{ref}(t) & = EB_{ref} + K + (\gamma_{FFM} + \delta) FFM_{ref} + (\gamma_{FM} + \delta) FM_{ref} + \dfrac{\eta_{FFM}}{\rho_{FFM}} (p\cdot EB_{ref} + g) \ & \qquad + \dfrac{\eta_{FM}}{\rho_{FM}} \big((1-p)\cdot EB_{ref} + g \big)
\end{aligned}
\end{equation}
represents the reference energy intake for normal growth. The energy balance for reference children is also of the form:
\begin{equation}
EB_{ref}(t) = A e^{-(t - t^{EB}{A})/\tau^{EB}{A}} + B e^{-(t - t^{EB}{B})^2/2(\tau^{EB}{B})^2} + D e^{-(t - t^{EB}{D})^2/2(\tau{D}^{EB})^2}.
\end{equation}
The reference fat free mass $FFM_{ref}$ and the reference fat mass $FM_{ref}$ in \eqref{iref} were obtained from linear interpolations to the values in table \ref{ref} which were obtained from [@ellis2000reference, @fomon1982body, @haschke1989body].
\begin{table}[!htb]
\centering
\begin{tabular}{lrrcrr}
& \multicolumn{2}{c}{\bf Males} & & \multicolumn{2}{c}{\bf Females} \
\cmidrule{2-3}
\cmidrule{5-6}
Age & Fat Mass (kg) & Fat Free Mass (kg) & & Fat Mass (kg) & Fat Free Mass (kg)\
\hline
\hline
4 & 14.0 & 2.7 & & 13.2 & 2.8 \
5 & 16.0 & 2.7 & & 14.7 & 2.9 \
6 & 17.9 & 2.8 & & 16.3 & 3.2 \
7 & 19.9 & 2.9 & & 18.2 & 3.7 \
8 & 22.0 & 3.3 & & 20.5 & 4.3 \
9 & 24.4 & 3.7 & & 23.3 & 5.2 \
10 & 27.5 & 4.8 & & 26.4 & 7.2 \
11 & 29.5 & 5.9 & & 28.5 & 8.5 \
12 & 33.2 & 6.7 & & 32.4 & 9.2 \
13 & 38.1 & 7.0 & & 36.1 & 10.0 \
14 & 43.6 & 7.2 & & 38.9 & 11.3 \
15 & 49.1 & 7.5 & & 40.7 & 12.8 \
16 & 54.0 & 8.0 & & 41.7 & 14.0 \
17 & 57.7 & 8.4 & & 42.3 & 14.3 \
18 & 60.0 & 8.8 & & 42.6 & 14.3 \
\hline
\hline
\end{tabular}
\caption{Reference values for fat and fat free mass (kg) from [@ellis2000reference, @fomon1982body, @haschke1989body] for interpolation to obtain $FM_{ref}$ and $FFM_{ref}$.}
\label{ref}
\end{table}
A closed form expression for the energy is given by: \begin{equation} E = \dfrac{K + (\gamma_{FFM} + \delta) FFM + (\gamma_{FM} + \delta) FM + \beta \cdot \Delta I + \Big( \frac{\eta_{FFM}}{\rho_{FFM}} p + \frac{\eta_{FM}}{\rho_{FM}}\cdot (1-p) \Big) \cdot EI + g \cdot \Big(\frac{\eta_{FFM}}{\rho_{FFM}} - \frac{\eta_{FM}}{\rho_{FM}} \Big)}{1 + \frac{\eta_{FFM}}{\rho_{FFM}} p + \frac{\eta_{FM}}{\rho_{FM}}\cdot (1-p) } \end{equation}
\subsubsection{Initial values of system of ordinary differential equations} To obtain the initial values for \eqref{diff} we estimated the initial fat mass using the equations from @deurenberg1991body: \begin{equation}\label{fm} FM_{0} = \begin{cases} 1.51 \cdot \text{BMI}_0 - 0.7 \cdot a - \frac{2.2}{100} \cdot BW_0 & \textrm{ if Male}\\ 1.51 \cdot BMI_0 - 0.7 \cdot a + \frac{1.4}{100} \cdot BW_0 & \textrm{ if Female} \end{cases} \end{equation} where $a$ represents the age (years); $\text{BMI}$ the initial body mass index ($\text{kg}/\text{m}^2$) and $BW_0$ the initial body weight of the child. The initial fat free mass is given by the difference between fat mass and body weight: \begin{equation}\label{ffm} FFM_0 = BW_0 - FM_0. \end{equation}
Solution algorithm
To solve this system of differential equations, we used a 4th order Runge-Kutta algorithm (RK4) @ascher2011first with a stepsize $\Delta t = 1$. RK4 was programmed in C++ for speed and connected to R via the Rcpp package [@Rcpp2, @Rcpp1].
Additional information
Additional information on the adult's model, and why to use the dynamic adult model instead of other classical approaches can be found in the package's vignettes:
browseVignettes("bw")
References
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