require("bw") require("ggplot2")
In this vignette we explain how to use the model for adults in R; we develop and explain the equations involved both for casual and advanced readers.
The main inputs for the body weight change model in adults are:
 Input  Meaning  Optional  Default 
::::
 bw
 Body Weight (kg)  No   
 ht
 Height (m)  No   
 age
 Age (yrs)  No   
 sex
 Either 'male'
or 'female'
 No   
As an example consider a 40
year old 'female'
weighting 80
kg with a height of 1.8
metres:
female_model1 < adult_weight(bw = 80, ht = 1.8, age = 40, sex = "female")
Furthermore, the model allows the user to input daily changes in sodium and energy intake compared to the baseline consumption (for example, if doing a diet which reduces 100
kcals from previous consumption).
 Input  Meaning  Optional  Default 
::::
 EIchange
 Energy intake difference from baseline (kcal)  Yes  No energy change 
 NAchange
 Sodium intake difference from baseline (kcal)  Yes  No sodium change 
Note that this 100
kcals is not cummulative. For example, if an individual consumes daily 2000
kcals a EIchange = 100
means that in day 1 the individual consumes 1900
kcals; in day 2 the individual consumes 1900
kcals; in day 3 the individual consumes 1900
kcals, etc.
For example, this female can reduce her energy consumption by 250
kcals and her sodium intake by 20
mg:
female_model2 < adult_weight(bw = 80, ht = 1.8, age = 40, sex = "female", EIchange = rep(250, 365), NAchange = rep(20, 365))
If no NAchange
is inputed, the model assumes a 0
change:
female_model3 < adult_weight(bw = 80, ht = 1.8, age = 40, sex = "female", EIchange = rep(250, 365))
The same is true for EIchange
.
Note that in the examples above, EIchange = rep(250, 365)
is inputed as a vector with each time step dt
(default, dt = 1
day) representing the consumption reduction for that day. See Energy Reduction section for additional information
Other (optional) inputs include:
 Input  Meaning  Optional  Default 
::::
 EI
 Energy intake (kcal) at baseline  Yes  Model estimate 
 fat
 Baseline fat mass (kg) of the individual  Yes  Model estimate 
 PAL
 Physical activity level  Yes  $1.5$ 
 pcarb_base
 \% of baseline intake that is carbohydrates  Yes  $0.5$ 
 pcarb
 \% of intake after day $0$ that is carbohydrates  Yes  pcarb_base

Hence the same female can consume 2000
kcals at baseline, with 22
kg of fat mass, having a physical activity level of 1.7
with a proportion of 0.6
kcals attributed to carbohydrates at baseline and a proportion of 0.5
kcals attributed to carbohydrates after dietary change.
female_model4 < adult_weight(bw = 80, ht = 1.8, age = 40, sex = "female", EIchange = rep(250, 365), EI = 2000, fat = 22, PAL = 1.7, pcarb_base = 0.6, pcarb = 0.5)
Additional inputs include days
to specify the model's time span and checkValues
which checks that the model's results are biologically plausible (for example, that modelled body weight is always positive). All inputs used in the model are:
 Input  Meaning  Optional  Default 
::::
 bw
 Body Weight (kg)  No   
 ht
 Height (m)  No   
 age
 Age (yrs)  No   
 sex
 Either 'male'
or 'female'
 No   
 EIchange
 Energy intake difference from baseline (kcal)  Yes  No energy change 
 NAchange
 Sodium intake difference from baseline (kcal)  Yes  No sodium change 
 EI
 Energy intake (kcal) at baseline  Yes  Model estimate 
 fat
 Fat mass (kg) of the individual  Yes  Model estimate 
 PAL
 Physical activity level  Yes  $1.5$ 
 pcarb_base
 \% of baseline intake that is carbohydrates  Yes  $0.5$ 
 pcarb
 \% of intake after day $0$ that is carbohydrates  Yes  pcarb_base

 days
 Time period (days) to run the model  Yes  $365$ 
 dt
 Time step for RungueKutta 4  Yes  $1$ 
 checkValues
 Check for internal consistency  Yes  TRUE

Finally, we remark that one can also input data from a database to estimate individuallevel weight change (see the related section
#Database information weights < c(45, 67, 58, 92, 81) heights < c(1.30, 1.73, 1.77, 1.92, 1.73) ages < c(45, 23, 66, 44, 23) sexes < c("male", "female", "female", "male", "male") #Returns a weight change matrix and other matrices database_model < adult_weight(weights, heights, ages, sexes)
Result plots can be obtained by model_plot
function:
model_plot(female_model2, "Body_Weight")
Plotting options include "Body_Weight"
, Fat_Mass
, Lean_Mass
, Glycogen
, Extracellular_Fluid
and Adaptive_Thermogenesis
. 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 intake change 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 by day 365
had reduced his consumption in 250
kcals and by day 730
had increased his consumption by 100
kcals (both compared to the consumption initially). The energy_build
function interpolates those values via a Brownian Bridge:
EIbrownian < energy_build(c(0, 250, 100), 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 change (kcals)") + ggtitle("Energy interpolation")
Such energy change matrix can be directly inputed in the model:
model_brownian < adult_weight(70, 1.75, 22, "male", EIbrownian, days = 730)
Other interpolation modes include Linear
, Exponential
, Stepwise_R
(right stepwise), Stepwise_L
(left stepwise), and Logarithmic
:
EIlinear < energy_build(c(0, 250, 100), c(0, 365, 730), "Linear") EIexponential < energy_build(c(0, 250, 100), c(0, 365, 730), "Exponential") EIstepwise_r < energy_build(c(0, 250, 100), c(0, 365, 730), "Stepwise_R") EIstepwise_l < energy_build(c(0, 250, 100), c(0, 365, 730), "Stepwise_L") EIlogarithmic < energy_build(c(0, 250, 100), 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 < adult_weight(70, 1.75, 22, "male", EIlinear, days = 730) model_exponential < adult_weight(70, 1.75, 22, "male", EIexponential, days = 730) model_logarithmic < adult_weight(70, 1.75, 22, "male", EIlogarithmic, days = 730) model_stepwise_r < adult_weight(70, 1.75, 22, "male", EIstepwise_r, days = 730) model_stepwise_l < adult_weight(70, 1.75, 22, "male", 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"))
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 in a for
loop). As an example consider the following dataset:
#Database information mydata < data.frame( id = 1:5, weight = c(67, 68, 69, 70, 71), height = c(1.30, 1.73, 1.77, 1.92, 1.73), age = c(45, 23, 66, 44, 23), sex = c("male", "female", "female", "male", "male"), energy_change = runif(5, 200, 200), prob = c(0.1, 0.2, 0.2, 0.05, 0.45)) #Get energy change with energy build function eichange < energy_build(cbind(rep(0,5), mydata$energy_change), c(0, 365)) #Returns a weight change matrix and other matrices database_model < adult_weight(mydata$weight, mydata$height, mydata$age, mydata$sex, eichange)
Plots can also be obtained for the population with the same command model_plot
:
model_plot(database_model, "Body_Weight")
Summary measures can be obtained via model_mean
and adult_bmi
. The first, quantifies mean for 'Body_Weight'
, 'Adaptive_Thermogenesis'
, 'Extracellular_Fluid'
, 'Age'
, 'Body_Mass_Index'
, 'Glycogen'
, 'Lean_Mass'
, and 'Fat_Mass'
:
model_mean(database_model, "Body_Weight")
head(model_mean(database_model, "Body_Weight"))[,1:5]
Mean is only estimated for some points in time, to estimate mean for the whole period, consider changing the days
vector variable:
model_mean(database_model, "Body_Weight", days = 1:365)
head(model_mean(database_model, "Body_Weight", days = 1:365))[,1:5]
Mean can also be grouped by a variable (say, sex
):
model_mean(database_model, "Body_Weight", days = 1:365, group = mydata$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$sex, design = design))[,1:5]
The same process can be replicated with adult_bmi
to obtain Body Mass Index prevalence per day:
adult_bmi(database_model, group = mydata$sex)
head(adult_bmi(database_model, group = mydata$sex))
Additional information on the Children model and other package functions can be obtained in the other package's Vignettes
browseVignettes("bw")
\noindent The main idea of the adult dynamic weight change model [@T7, @T8, @T9, @T10] is that body weight $BW(t)$ at a given time $t$ is the sum of four components: lean mass $L(t)$, fat mass $F(t)$, glycogen $g(t)$, and extracellular fluid $ECF(t)$:
\begin{equation}\label{bw} \overbrace{BW(t)}^{\textrm{Weight}} = \underbrace{F(t)}{\textrm{Fat}} + \overbrace{L(t)}^{\textrm{Lean}} + \underbrace{3.7 \cdot G(t)}{\textrm{Glycogen and Associated water}} + \overbrace{ECF(t)}^{\textrm{Extracellular Fluid}} \end{equation}
Lean mass, fat mass, and glycogen are interrelated, meanwhile the extracellular fluid has its own dynamic. The dynamic model's equations are congruent with the energy conservation law. Changes in weight are therefore determined by:
Changes in lean and fat mass are given by changes in intake, expenditure, and glycogen. An increase or decrease in body weight depends on the difference between intake and expenditure. A fraction of the change in weight $p$ will correspond to the change in lean mass and the remaining fraction $(1p)$ will correspond to changes in fat mass. The corresponding equations are:
\begin{equation}\nonumber \overbrace{\dfrac{dL}{dt}}^{\textrm{Change in lean mass}} = \underbrace{p}_{\textrm{Fraction}} \overbrace{\frac{\Big( EI  EE  \rho_G \frac{dG}{dt}\Big)}{\rho_L}}^{\textrm{Energy Intake}  \textrm{Expenditure+Storage}} \end{equation}
\begin{equation}\nonumber \overbrace{\dfrac{dF}{dt}}^{\textrm{Change in fat mass}} = \underbrace{(1p)}_{\textrm{Remaining fraction}} \overbrace{\frac{\Big( EI  EE  \rho_G \frac{dG}{dt}\Big)}{\rho_L}}^{\textrm{Energy Intake}  \textrm{Expenditure+Storage}} \end{equation}
Changes in glycogen are determined by the proportion of energy consumption from carbohydrates $CI$:
\begin{equation} \overbrace{\dfrac{dG}{dt}}^{\textrm{Change in glycogen}} = \underbrace{\frac{1}{\rho_G} \Big( CI  k_G G^2\Big)}_{\textrm{Carbohydrates and constant parameters}}, \end{equation}
Finally, the equation for extracellular fluid depends on sodium intake and carbohydrates consumed: \begin{equation}\nonumber \overbrace{\dfrac{dECF}{dt}}^{\textrm{Change in fluid}} = \overbrace{\frac{ \Big( \Delta Na_{diet} + \xi_{Na}(ECF  ECF_{init})  \xi_{CI} (1  \frac{CI}{CI_{b}}) \Big)}{Na}.}^{\textrm{Depends on changes in sodium and carbohydrat intake.}} \end{equation}
To summarize: weight is divided into four interdependent components: lean mass, fat mass, glycogen, and extracellular fluid. The weight change model describes the processes resulting from interconnections between the components:
\begin{equation}\label{bw2} \overbrace{BW(t)}^{\textrm{Weight}} = \underbrace{F(t)}{\textrm{Fat}} + \overbrace{L(t)}^{\textrm{Lean}} + \underbrace{3.7 \cdot G(t)}{\textrm{Glycogen and Associated water}} + \overbrace{ECF(t)}^{\textrm{Extracellular Fluid}} \end{equation}
The dynamic weight change model @T7 defines individual weight (kg) as the sum of fat $F$ and lean mass $L$, extracellular fluid $ECF$ and glycogen $G$: \begin{equation}\label{bodyweight} \begin{aligned} BW = ECF + 3.7 G + F + L. \end{aligned} \end{equation} Extracellular fluid $ECF \equiv ECF(t)$ is the solution to the ordinary differential equation system: \begin{equation}\label{ecf} \begin{cases} \dfrac{dECF}{dt} & = \frac{1}{Na} \Big( \Delta Na_{diet} + \xi_{Na}(ECF  ECF_{init})  \xi_{CI} (1  CI /CI_{b}) \Big),\\ ECF_{init} & = 0. \end{cases} \end{equation} where $Na = 3.22$ mg/ml, $\xi_{Na} = 3000$ mg/L/d, and $\xi_{CI} = 4000$ mg/d, are phisiological constants [@T7]. $\Delta Na_{diet}$ represents the change in sodium (mg/d) for the individual. $CI_b$ is the carbohydrate intake at baseline (assumed to be $1/2$ of energy intake estimated) and $CI \equiv CI(t)$ is the carbohydrate intake after the consumption reduction (assumed to be half the energy intake after reduction, $CI \equiv \Delta EI(t) / 2$). Glycogen mass $G \equiv G(t)$ is described by the ordinary differential equation: \begin{equation}\label{gly} \dfrac{dG}{dt} = \frac{1}{\rho_G} \Big( CI  k_G \cdot G^2\Big), \end{equation} where $\rho_G = 4206.501 \textrm{kcals}/\textrm{kg}$ ($17.6 MJ/kg$), and $k_G = CI_b/G_{init}^2$ is a constant with $G_{init} = 0.5$ kg the initial glycogen mass.
Fat and lean mass, $F\equiv F(t)$ and $L \equiv L(t)$, represent the solutions to the following system of nonlinear ordinary differential equations: \begin{equation}\label{desolveadoras} \begin{aligned} \dfrac{dF}{dt} & = \frac{\big(1p \big)}{\rho_F}\Big( EI_{\textrm{init}}  EE  \rho_G \frac{dG}{dt}\Big),\ \dfrac{dL}{dt} & = \frac{p}{\rho_L}\Big( EI_{\textrm{init}}  EE  \rho_G \frac{dG}{dt}\Big), \end{aligned} \end{equation} with $\rho_F = 9440.727$ kcals/kg ($39.5$ MJ/kg), $\rho_L = 1816.444$ kcals/kg ($7.6$ MJ/kg) are constants, and $p \equiv C/ (C+F)$ a function of fat mass with $C = 10.4 \cdot \rho_L \rho_F^{1}$. Total energy expenditure $EE$ is given by: \begin{equation} EE = K + \gamma_F F + \gamma_L L + \delta BW + TEF + AT + \eta_L \dfrac{dL}{dt} + \eta_F \dfrac{dF}{dt} , \end{equation} with $\gamma_F = 3.107075$ kcals/kg/d ($13$ kJ/kg/d), $\gamma_L = 21.98853$ kcals/kg/d ($92$ kJ/kg/d), $\eta_F = 179.2543$ kcals/kg ($750$ kJ/d), $\eta_L = 229.4455$ kcals/kg ($960$ kJ/kg) are physiological constants. $K$ is determined by the initial energy balance condition: \begin{equation}\label{K} K = RMR_{init} \cdot PAL  \gamma_L L_{init}  \gamma_F F_{init}  \delta BW_{init} \end{equation} with $RMR_{init}$ the initial resting metabolic rate, $PAL$ the physical activity level (assumed $PAL = 1.5$), $L_{init}, F_{init}, BW_{init}$ the initial lean, fat and body weight masses. The constant $\delta$ is determined defined as $\delta = RMR_{init} \Big( (1  \beta_{TEF}) \cdot PAL  1\Big) / BW_{init}$ with $\beta_{TEF} = 0.1$. Furthermore, the thermal effect of feeding is defined as $TEF \equiv \beta_{TEF} \Delta EI(t)$ with $\Delta EI(t)$ as previously specified. Finally, adaptative thermogenesis is given by the solution to the ODE system: \begin{equation}\label{at} \begin{cases} \dfrac{dAT}{dt} & = \beta_{AT} \Delta EI  AT, \ \ AT_{init} & = \beta_{AT} \cdot PAL \cdot RMR_{init}. \end{cases} \end{equation} We remark that for each individual, the initial resting metabolic rate $RMR_{init}$ is described by the equations @mifflin1990new: \begin{equation}\label{rmr} RMR_{init} = \begin{cases} 9.99\cdot BW_{init} + 625\cdot H_{init} 4.92\cdot AGE_{init} + 5 & \text{if Sex $=$ Male}, \ 9.99\cdot BW_{init} + 625\cdot H_{init}4.92\cdot \cdot AGE_{init} 161 & \text{if Sex $=$ Female}. \end{cases} \end{equation} with $H_{init}$, $AGE_{init}$ initial height and age respectively. Initial fat mass was obtained via the function: \begin{equation}\label{fatmass} F_{init} = \begin{cases} (1/100) \cdot BW_{init} \cdot \Big(0.14 \cdot AGE_{init} + 37.31 \times \ln(BW_{init} / H_{init}^2)  103.94 \Big) & \text{if Sex $=$ Male} , \ (1/100) \cdot BW_{init} \cdot \Big(0.14 \cdot AGE_{init} + 39.96 \cdot \ln(BW_{init} / H_{init}^2)  102.01 \Big) & \text{if Sex $=$ Female}. \end{cases} \end{equation}
Additional information on the model can be found in [@T7, @T8, @T9, @T10]
To solve this system of differential equations, we used a 4th order RungeKutta 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 on the children'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")
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