Human PBPK model for dioxin

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Question

How to calculate dioxin concentrations in human tissues after a given exposure pattern?

Answer

These are several approaches available.

INERIS (France) has produced a TOXI/INERIS model.
Aylward (2005a, b) have described a model.^[1]^[2]

Tuomisto et al model

This model was previously on page Dioxin.

Dioxin parameters(%,-,d-1,a)
Obs	Congener	Market basket	TEF	Elimination constant	Half life
1	TCDD	2	1	0.00026	7.3
2	12378PeCDD	2	1	0.00017	11.2
3	123678HxCDD	5	0.1	0.000145	13.1
4	1234678HpCDD	7	0.001	0.00039	4.9
5	OCDD	60	0.0003	0.00028	6.8
6	TCDF	8	0.1	0.0009	2.1
7	23478PeCDF	16	0.3	0.00027	7.0

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library(OpasnetUtils)
library(ggplot2)

####### FETCH AND MODIFY INPUTS

parameters <- opbase.data("Op_en1947.dioxin_parameters")
parameters$Obs <- NULL

congeners <- Ovariable("congeners", data = data.frame(Congener = congeners, Result = 1))

TEF 	<- Ovariable("TEF", 
	data = parameters[parameters$Observation == "TEF" ,  colnames(parameters) != "Observation"]
)

share	<- Ovariable("share", 
	data = parameters[parameters$Observation == "Market basket" , colnames(parameters) != "Observation"]
)
share <- share * TEF * congeners
share <- share / oapply(share, cols = "Congener", FUN = sum)

dose <- dose * 365 * share # From days to years and congener-specific doses

ke	<- Ovariable("ke", 
	data = parameters[parameters$Observation == "Elimination constant" ,  colnames(parameters) != "Observation"]
)
ke <- ke * 365 # from days to year

time	<- Ovariable("time", data = data.frame(Time = 0:50, Result = 0:50))

concentration <- Ovariable("concentration", 
	dependencies = data.frame(Name = c("ke", "dose", "weight.increase", "fat", "time")),
	formula = function(...) 
	{
		out <- dose / ke * # Dioxin amount in steady state
			(1 - exp(-1 * ke * time)) / # Fraction of steady state reached at time 'time'.
			(fat + weight.increase * time) # Amount converted to fat concentration.

		return(out)
	}
)

concentration <- EvalOutput(concentration)

ggplot(concentration@output, aes(x = timeResult, y = concentrationResult, colour = Congener)) + geom_line() + 
	theme_gray(base_size = 28) + labs(y = "Concentration (pg /g fat)", x = "Years of exposure")

Dioxin kinetics with Bayes

This code works only if you first get the data.frame dat. It was originally produced with the code KTL Sarcoma study#Data management. Data files from the study must be available to produce dat.

Correlations from the sarcoma study data
	WHOTEQ	mu
Pearson
Observed WHOTEQ	1.000000	0.380774
Predicted mu	0.380774	1.000000
Spearman
WHOTEQ	1.0000000	0.5486548
mu	0.5486548	1.0000000

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##################################### JAGS

library(rjags)

### INITIATE

prediction <- function(dat, est) {
	out <- list()
	out$crosscorrelation <- crosscorr(est)
	out$crosscorrplot <- crosscorr.plot(est)
	est <- as.data.frame(est[[1]])
	k <- est$k
	krint <- est$krint
#	muutosk <- est$muutosk
		teq <- dat$WHOTEQ
#		sil <- dat$Silmaara
		dose <- dat$Dioxin * 365
		time <- dat$ika - 15
		fat <- dat$Paino^2 / dat$Pituus^2 * 1E+5 # paino(g) * BMI/100
		trint <- dat$Rintamaitoa # Rintaruokinta-aika yhteensä (kk)
		tsynt <- dat$Syntymasta # Aika viimeisen lapsen syntymästä (a)
		Pnetto <- dat$Pnetto # Painon muutos (kg) viiden vuoden aikana
	weight.increase <- 0
	m <- est$m
	r <- 0.03
	i <- 1:nrow(dat)
	dat$mu <- (dose[i] / (r - k) * # Dioxin amount in steady state (although it is not steady of r != 0)
		(exp((r - k) * time[i]) - 1) * # Fraction of steady state reached during the period.
		(1 - (1 - exp(-krint * trint[i])) * exp(-k * tsynt[i])) + # Impact of breast feeding
#		muutosk * Pnetto[i] + # A test for the impact of weight change in 5 years.
		m * exp(-1 * k * time[i])) / # Residual amount from the beginning.
		(fat[i] + weight.increase * time[i]) # Amount converted to fat concentration.

	out$plot <- ggplot(dat, aes(x = WHOTEQ, y = mu)) + geom_point() + 
		geom_line(data = data.frame(D = c(0, 150)), aes(x = D, y = D)) + geom_smooth()
	out$Pearson <- cor(dat[c("WHOTEQ", "mu")])
	out$Spearman <- cor(dat[c("WHOTEQ", "mu")], method = "spearman")

	return(out)
}

##### PREPROCESS DATA

dat <- dat[
	!is.na(dat$WHOTEQ) &
	!is.na(dat$mSilakkaa) &
	!is.na(dat$Kalteq) &
	!is.na(dat$Paino) &
	!is.na(dat$Pituus) &
	!is.na(dat$Pnetto) &
	!is.na(dat$ika) , 
]
N <- nrow(dat)

mo <- textConnection("model {
	for (i in 1:N) {
		teq[i] ~ dnorm(mu[i], taut)
		mu[i] <- (dose[i] / (r - k) * # Dioxin amount in steady state (although it is not steady of r != 0)
			(exp((r - k) * time[i]) - 1) * # Fraction of steady state reached during the period.
			(1 - (1 - exp(-krint * trint[i])) * exp(-k * tsynt[i])) + # Impact of breast feeding
#			muutosk * Pnetto[i] + # A test for the impact of weight change in 5 years.
			m * exp(-1 * k * time[i])) / # Residual amount from the beginning.
			(fat[i] + weight.increase * time[i]) # Amount converted to fat concentration.

	}
#	m ~ dnorm(0, 0.0001)
m <- 0
	r <- 0.03
	taut <- pow(sigmat, -2)
	sigmat ~ dunif(0, 100)
	weight.increase <- 0
	k ~ dunif(0.03, 0.15)
	krint ~ dunif(0.01, 0.08)
#	muutosk ~ dnorm(0, 0.0001)
}")

jags <- jags.model(mo,
	data = list(
		teq = dat$WHOTEQ,
#		sil = dat$Silmaara,
		dose = dat$Dioxin * 365,
		time = dat$ika - 15,
		fat = dat$Paino^2 / dat$Pituus^2 * 1E+5, # paino(g) * BMI/100
		trint = dat$Rintamaitoa, # Rintaruokinta-aika yhteensä (kk)
		tsynt = dat$Syntymasta, # Aika viimeisen lapsen syntymästä (a)
#		Pnetto = dat$Pnetto, # Painon muutos (kg) viiden vuoden aikana
		N = N),
	n.chains = 4,
	n.adapt = 100
)

close(mo)
#update(jags, 1000)

out <- coda.samples(jags, c('m', 'taut', 'k', 'krint'), nrow(dat)) # Stores a posterior sample
plot(out)

pred <- prediction(dat, out)

cat("Crosscorrelation\n")
print(pred$crosscorrelation)
cat("Crosscorrelation plot\n")
print(pred$crosscorrplot)
cat("Observed and predicted concentrations\n")
print(pred$plot + theme_gray(base_size = 24) + labs(
	title = "WHOTEq concentrations in humans", 
	x = "Observed (pg/g fat)",
	y = "Predicted (pg/g fat)"
))
cat("Pearson\n")
print(pred$Pearson)
cat("Spearman\n")
print(pred$Spearman)

Aylward et al model

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library(reshape2)

##### f.h a function for fraction of TCDD in liver. Parameters:
#     C.b Concentration of TCDD in body (ng/kg)
#     f.h.min Minimum proportion of body burden distributed to liver (unitless)
#     f.h.max Maximum proportion of body burden distributed to liver (unitless)
#     K Body concentration for half-maximum increase in liver distribution (ng/kg)
f.h <- function(C.b, f.h.min = 0.01, f.h.max = 0.7, K = 100){
out <- f.h.min + (f.h.max - f.h.min) * C.b / (K + C.b)
return(out)
}
#####


##### C.bt  a function for calculating body concentration of TCDD. Parameters:
#     C.b   Body concentration of TCDD at time point t
#     time  length of follow-up time (a). May also be negative.
#     beta  Dose rate or amount of TCDD per one exposure unit (ng/unit)
#     E.t   Exposure measured in exposure units (unit/a)
#     BW    Body weight (kg)
#     k.e   Hepatic elimination (1/a)
#     k.a   Adipose elimination (1/a)
#     C.at  Adipose concentration of TCDD (ng/kg)
#     w.a   Fraction of body weight in adipose tissue (unitless)
#     w.h   Fraction of body weight in liver (unitless)

C.bt <- function(C.b, time = 10, E.t = 1, beta = 134, BW = 70, k.a = 0.01, k.e0 = 0.85, k.emin = 0.2, BMI = 24, age = 30, sex = 1, w.h = 0.03){
t    <- (0:(time*12))/12
dt   <- time/abs(time)/12
out  <- data.frame(Time = t, w.a = NA, C.a = NA, C.b = NA, C.h = NA, k.e = NA)
for(i in 1:length(t)){
f.he <- f.h(C.b)
k.e  <- max(k.e0 - 0.011 * age, k.emin)
w.a  <- (1.2 * BMI + 0.23 * age - 10.8*sex - 5.4)/100
C.a  <- C.b * (1 - f.he) / w.a
C.h  <- C.b * f.he /  w.h
out[i, 2:ncol(out)] <- c(w.a, C.a, C.b, C.h, k.e)
C.b  <- C.b + beta * E.t * dt / BW - k.e * dt * f.he * C.b - k.a * dt * C.a * w.a
age  <- age + dt
}
return(out)
}


out <- C.bt(C.b, E.t = E.t, time = time)
head(out)
out <- melt(out, id.vars = "Time")
colnames(out) <- c("DateTime", "TagName", "Value")
out <- dropall(out[out$TagName %in% select, ])

cat("Figure 1. Variables as functions of time. C.b: body burden of TCDD, C.h and C.a: liver and fat concentrations of TCDD, respectively, k.e elimination rate, a.w fraction of body weight in fat.\n")
multivarplot(out)
C.b <- out[out$TagName == "C.b", "Value"]
C.b <- min(C.b):max(C.b)
cat("Figure 2. Fraction of body TCDD in liver as a function of body burden.\n")
plot(C.b, f.h(C.b))

TOXI/INERIS model

Dioxin PBPK model
The results are total amounts (ng) of dioxin, except blood concentration (ng/l). See the model documentation
The input data used for this variable: Intake at age 0 - 5 a (ng/min) 0.000173611 Intake at age 5 - 10 a (ng/min) 0.000173611 Intake at age 10-15 a (ng/min) 0.000173611 Intake at age 15 - 40 a (ng/min) 0.000173611 Intake at age 40 - a (ng/min) 0.000173611 Start of peak intake (days) 100 End of peak intake (days) 1000 Additional peak intake (ng/min) 0.000001
Click here to run the model

Rationale

Tuomisto et al

This model is based on one-compartment kinetics with a dioxin intake trend (r %/a decrease). t_i means the distance in time between timepoint i (in the history) and now. The amount of dioxin in the body now (m_n) from a dose d_i at timepoint i is calculated as follows, if the dose during time decreases at a constant rate r:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_n = d_i e^{-k t_i} = d_n e^{r t_i} e^{-k t_i} = d_n e^{(r-k) t_i}.}

The total amount of dioxin M_n from all doses during the time period is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n = \sum_{i=1}^n d_n e^{(r-k) t_i} = \int_0^n d_n e^{(r-k) t}dt = d_n \frac{e^{(r-k) t}}{r-k} - d_n \frac{e^{(r-k) 0}}{r-k} = \frac{d_n}{r-k} (e^{(r-k) t}-1).}

There may also be a single additional dioxin load at timepoint A, and this is denoted m_A:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n = \frac{d_n}{r-k} (e^{(r-k) t}-1) + m_A e^{-k t_A}.}

Also, there may be a breast feeding period during the observed time span

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n = (\frac{d_n}{r-k} (e^{(r-k) t}-1) + m_A e^{-k t_A}) (1 - (1-e^{-k_B t_B})(e^{-k t_S})),}

where k_B is the elimination constant caused by breast feeding, t_B is the duration of breast feeding, and t_S is the time since the breast feeding.

If the intake is constant and there is no breast feeding, this simplifies into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_n = \frac{d_n}{k} (1 - e^{-k t}) + m_A e^{-k t_A},}

where d_n/k is the steady state amount at constant intake, (1 - e^{-k t}) is the relative deviation from the steady state, and m_A e^{-k t_A} is the burden from the additional dioxin load at A.

Assumptions

WHO-TEQ intake 57 pg/d average (Kiviranta et al, Env Int 2004:30:923)
- low intake 40 pg/d (42,05)
- high intake 160 pg/d (168,2)
Average distribution 56 pg/d animal products, 1 pg/d plant products (ibid)
- low intake 39 pg/d animal products, 1 pg/d plant products
- high intake 157 pg/d animal products, 3 pg/d plant products
Congener distribution as pg TEQ (rounded)
- low intake TCDD 7,9, 12378PCDD 7,9, 123678HxCDD 2, 1234678HpCDD 0,03, OCDD 0,07, TCDF 3,2, 23478PeCDF 19 pg/d WHO-TEQ
- high intake TCDD 32, 12378PCDD 32, 123678HxCDD 8, 1234678HpCDD 0,1, OCDD 0,3, TCDF 12,6, 23478PeCDF 76 pg/d WHO-TEQ
Congener distribution as pg (rounded, rounded numbers used in other sheets)
- low intake TCDD 7,9, 12378PCDD 7,9, 123678HxCDD 20, 1234678HpCDD 28, OCDD 237, TCDF 32, 23478PeCDF 63 pg/d
- high intake TCDD 32, 12378PCDD 32, 123678HxCDD 80, 1234678HpCDD 111, OCDD 948, TCDF 126, 23478PeCDF 253 pg/d
Congener elimination constants (d-1) TCDD 0,00026, 12378PCDD 0,00017, 123678HxCDD 0,000145, 1234678HpCDD 0,00039, OCDD 0,00028, TCDF 0,0009, 23478PeCDF 0,00027
- Half lives from Milbrath et al., Environ. Health Persp. 2009:117:417-425

TOXI/INERIS

This model description has been taken from an equivalent page of the IEHIAS-project. For the main article about dioxins, see Dioxin.

Basic info

Last modification:	04/07/2007	Model version :
Software status :	Free software	OS :	Linux
Supplier :	INERIS	Installation :	See source
Possible developments :	yes	Source :	TOXI/INERIS web site ^[3]
Supplier address:	INERIS Verneuil en Halatte, France	Referent(s)* :	S. MICALLEF (INERIS) Sandrine.micallef@ineris.fr

Discipline : keywords: toxicokinetic model, 2,3,7,8-tétra-chloro-p-dioxin (TCDD), PBPK

Scope of the mode

Physiologically Based Toxicokinetic (PBTK) model for Dioxine. The presented PBTK model allows to simulate ingestion exposures to 2,3,7,8-tétra-chloro-p-dioxin (TCDD) for a woman over her whole life. TCDD is a persistent chemical found in trace amounts all over the globe. It accumulates in animal fat all along the trophic chain. Human exposure to TCDD is therefore almost unavoidable, even if in trace amounts. TCDD has multiple effects on health.

Model description

The proposed model is based on a previous one proposed by van der Molen and colleagues in 1996^[4]. The model computes various measures of internal dose as a function of time. The superposition of a peak exposure to a time-varying background intake can be described. All ingested TCDD is supposed to be absorbed. TCDD is supposed to distribute between blood, fat, muscles and skin, and viscera. The model equations are solved dynamically (by numerical integration) with the MCSim simulation package to give a good precision both on short-term and long-term scales. The body mass and the volume of the various body tissues change with the age of the simulated individual.

Figure 1 : Toxicokinetic model used to describe TCDD toxicokinetic in the human body^[4]. compartments are characterized with volume V and partition coefficient P. Exchanges between are governed by blood flows, F. Elimination is assumed proportional to the elimination constant, ke. The set value of each parameter is given in Table 1. The ingested quantity by unit of time ki (in ng/min), is determined by the exposure scenario.

Figure 1 gives a graphical representation of the model used. Only ingestion exposure is described in this model (the totality of the exposure dose is assumed to be absorbed). The TCDD is supposed to be distributed into different compartments of the body : blood, fat, muscles and skin. The original formulation of van der Molen and coll.^[4] regards all these compartments as being with balance in an instantaneous way. This assumption is acceptable only if slow evolutions of absorption are the limiting factor of the kinetics of the product. Since the simulation of a short peak of exposure interests us, we developed a traditional dynamic formulation^[5], specifies at the same time on short scales of time and the long-term.

Model equations Equations defining the proposed model are the following : For quantites of TCDD in fat, viscera, muscle and skin, and liver : (1) (2) (3) (4) La concentration artérielle est calculée par : (5)

The cardiac output, Ft, is proportinal to the ventilation rate, Fp : (1) The body volume evolve as a function of age : (6) The volume of fat, viscera, liver also evolve as a function of age^[4]: (7) (8) (9) Volume of "muscles and skin" compartment is calculated as the difference between 90% of the total body volume (because bones are not included) and the other compartments : (10) Units used are : quantities of TCDD are expressed in ng, TCDD concentrations in ng/L, age in years volumes in liter, flows in L/min, the elimination constant in min-1, the ingested quantity by unit of time in ng/min. The body density is assumed equal to 1.

Figure 2 presents the temporal evolution of these parameters for a woman (the evolution is overall similar for a man). The reference averaged values used for the parameters not evolving with time are given in Table 1. The equations of the model were coded using the MCSim software^[6].

Figure 2 : Temporal evolution of volumes for the woman^[4].

Applications examples

Example of the use of this model can be found in the paper by Bois^[7]

Model inputs

All intakes are given in ng/min. A typical unit is pg/d; to change from the latter to the former, divide by 1440000.

Background exposure parameters:

dose1: intake of dioxin during age 0 - 5 years
dose2: intake of dioxin during age 5 - 10 years
dose3: intake of dioxin during age 10 -15 years
dose4: intake of dioxin during age 15 - 40 years
dose5: intake of dioxin during age 40 - years

Peak exposure parameters (in addition to the background)

peakstart: start of the peak exposure period (in days of age)
peakend: end of the peak exposure period (in days of age)
peakdose: Additional intake of dioxin during the peak period (in ng/min)

This model relates to physiologically based toxicokinetics modelling process. The presented PBTK model allows to simulate ingestion exposures to 2,3,7,8-tétra-chloro-p-dioxin (TCDD) for a woman over her whole life. The model computes various measures of internal dose as a function of time. The superposition of a peak exposure to a time-varying background intake can be described. The model equations are solved dynamically (by numerical integration) with the MCSim simulation package to give a good precision both on short-term and long-term scales. The body mass and the volume of the various body tissues change with the age of the simulated individual.

Model description

Purpose

This model relates to physiologically based toxicokinetics modelling process. The presented PBTK model allows to simulate ingestion exposures to 2,3,7,8-tétra-chloro-p-dioxin (TCDD) for a woman over her whole life. TCDD is a persistent chemical found in trace amounts all over the globe. It accumulates in animal fat all along the trophic chain. Human exposure to TCDD is therefore almost unavoidable, even if in trace amounts. TCDD has multiple effects on health.

Boundaries

It is related to woman exposure through food.

Input

Input data have to be written directly on the webpage. Data consist in: background rates, for different periods of life, beginning, end and level of peak exposure.

Output

Output data are the concentrations in different parts of the body (blood, liver, fat, well-perfused tissues and poorly-perfused tissues) as a function of time.

Description of processes modelled and of technical details

Figure 1 : Toxicokinetic model used to describe TCDD toxicokinetic in the human body (d'après (Van der Molen et al. 1996)). compartments are characterized with volume V and partition coefficient P. Exchanges between are governed by blood flows, F. Elimination is assumed proportional to the elimination constant, ke. The set value of each parameter is given in Table 1. The ingested quantity by unit of time ki (en ng/min), is determined by the exposure scenario.

**Table 1 : Numerical values of the parameters of the toxicokinetic model for TCDD in the woman.**
Parameter^(a)	Symbol	Numerical Value
Ventilation rate	F_p	8,0
Blood over air ventilation rate	R	1,14
Blood flow rates
Fat	F_f	0,09
Liver	F_l	0,24
Muscles and skin	f_m	0,18
Viscera	f_v	– ^(b)
Volumes
Total body volume	V_t	– ^(c)
Fat	V_f	– ^(c)
Liver	V_l	– ^(c)
Muscles and skin	V_m	– ^(c)
Viscera	V_v	– ^(c)
Partition Coefficient
Fat	P_f	300
Liver	P_l	25
Muscles and skin	P_m	4
Viscera	P_v	10
Elimination constant	k_e	8,45´;10^{-8 (d)}

(a) Units : volumes (L), blood flow (L/min), et elimination constant (min-1).

(b) Blood flow rate to viscera is calculated by difference between 1 and the sum of blood flow rates toward the other compartments.

(c) Volumes evolve with time.

(d) Corresponds to a half-life of 15,6 years.

Figure 1 gives a graphical representation of the model used. Only ingestion exposure is described in this model (the totality of the exposure dose is assumed to be absorbed). The TCDD is supposed to be distributed into different compartments of the body : blood, fat, muscles and skin. The original formulation of van der Molen and coll. (Van der Mollen et al. 1996) regards all these compartments as being with balance in an instantaneous way. This assumption is acceptable only if slow evolutions of absorption are the limiting factor of the kinetics of the product. Since the simulation of a short peak of exposure interests us, we developed a traditional dynamic formulation (Gerlowski et al. 1983), specifies at the same time on short scales of time and the long-term.

Model equations

Equations defining the proposed model are the following:

For quantities of TCDD in fat, viscera, muscle and skin, and liver:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta Q_f}{\delta t} = f_f \cdot F_t \Bigl( C_{art} - \frac{Q_f}{V_f \times P_f} \Bigr) \qquad (1) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta Q_v}{\delta t} = f_v \cdot F_t \Bigl( C_{art} - \frac{Q_v}{V_v\times P_v} \Bigr) \qquad (2) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta Q_m}{\delta t} = f_m \cdot F_t \Bigl( C_{art} - \frac{Q_m}{V_m\times P_m} \Bigr) \qquad (3) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\delta Q_l}{\delta t} = f_l \cdot F_t \Bigl( C_{art} - \frac{Q_l}{V_l\times P_l} \Bigr) - k_e \times Q_l + k_i \qquad (4) }

The arterial concentration is calculated by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{art} = \Bigl(\frac{f_f \times Q_f}{V_f \times P_f} + \frac{f_v \times Q_v}{V_v \times P_v} + \frac{f_m \times Q_m}{V_m \times P_m} + \frac{f_f \times Q_l}{V_f \times P_l}\Bigr) \qquad (5) }

Cardiac output F_t is proportional to the ventilation rate Fp:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_t = 0.7 \times F_p \times R \qquad (6) }

The body volume evolves as a function of age:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_t = 0.1959 \times t - \frac{57.497}{(1+4.617 \times exp(-0.572 \times (t-11.33)))^{1/4.617}} \qquad (7) }

The volume of fat, viscera, liver also evolve as a function of age (Van der Molen et al. 1996):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_g = \begin{cases} 0.5+4.5 \times \frac{t}{10}, \qquad\text{if }t\ge 10\text{,}\\ 0.5+4.5+8\times\frac{t-10}{15}, \qquad\text{if }10 \le t \le 15\\0.5+4.5+8+17\times\frac{t-15}{55}, \qquad\text{if }t \ge15\text{.}\end{cases} \qquad (8) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_v = \frac{6.095}{(1+4.617 \times exp(-0.3937 \times (t-6.5582)))^{1/4.617}} \qquad (9) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_f = \frac{1.758}{(1+4.617 \times exp(-0.3309 \times (t-12.478)))^{1/4.617}} \qquad (10) }

Volume of "muscles and skin" compartment is calculated as the difference between 90% of the total body volume (because bones are not included) and the other compartments:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_m = 0.9 \times V_t - V_f - V_v - V_l \qquad (11) }

Units used are : quantities of TCDD are expressed in ng, TCDD concentrations in ng/L, age in years volumes in liter, flows in L/min, the elimination constant in min-1, the ingested quantity by unit of time in ng/min. The body density is assumed equal to 1.

Figure 2 presents the temporal evolution of these parameters for a woman (the evolution is overall similar for a man). The reference averaged values used for the parameters not evolving with time are given in Table 1. The equations of the model were coded using the MCSim software (Bois et al. 1997).

Figure 2 : Temporal evolution of volumes for the woman (Van der Molen et al. 1996).

Rationale

The proposed model is based on a previous one proposed by van der Mollen and colleagues in 1996 (Van der Mollen et al. 1996). The model computes various measures of internal dose as a function of time. The superposition of a peak exposure to a time-varying background intake can be described. All ingested TCDD is supposed to be absorbed. TCDD is supposed to distribute between blood, fat, muscles and skin, and viscera. The model equations are solved dynamically (by numerical integration) with the MCSim simulation package to give a good precision both on short-term and long-term scales. The body mass and the volume of the various body tissues change with the age of the simulated individual.

See also

Contact person: Frédéric Y. Bois (frederic.bois [at] ineris.fr
Time required for a typical run: Typically one second
Degree of mastery: Basic user / Expert user
Intellectual property rights (who is allowed to access or use the model?): Free software.
Tool website: [2] and [3]

References

↑ Aylward LL et al. (2005a) Concentration-dependent TCDD eliminationi kinetics in humans: toxicokinetic modeling for moderately to highly exposed adults from Seveso, Italy, and Vienna, Austria, and impact on dose estimates for the NIOSH cohort. Journal of Exposure Analysis and Environmental Epidemiology 15: 51-65.
↑ Aylward LL et al. (200b) Exposure reconstruction for the TCDD-exposed NIOSH cohort using a concentration- and age-dependent model of elimination. Risk Analysis 25: 4: 945-956.
↑ http://toxi.ineris.fr/activites/toxicologie_quantitative/toxicocinetique/modeles/dioxine/sub_dioxine.php
↑ ^4.0 ^4.1 ^4.2 ^4.3 ^4.4 Van der Molen, G. W., S. A. L. M. Kooijman and W. Slob (1996). "A generic toxicokinetic model for persistent lipophilic compounds in humans: an application to TCDD." Fundamental and Applied Toxicology 31: 83-94.
↑ Gerlowski, L. E. and R. K. Jain (1983). "Physiologically based pharmacokinetic modeling: principles and applications." Journal of Pharmaceutical Sciences 72: 1103-1127.
↑ Bois, F. Y. and D. Maszle (1997). "MCSim: a simulation program." Journal of Statistical Software 2(9): [1].
↑ Bois, F. Y. (2003). "Modélisation toxicocinétique de la concentration sanguine de 2,3,7,8-tetrachloro-p-dioxine après ingestion chez la femme." Environnement, Risques et Santé 2(1). [Toxicokinetic modelling of 2,3,7,8-tétrachloro-p-dioxin blood concentration after ingestion by women. Environnement, Risque et Santé, (2003) 2:45-53. ]

Bois, F. Y. (2003). "Modélisation toxicocinétique de la concentration sanguine de 2,3,7,8-tetrachloro-p-dioxine après ingestion chez la femme." Environnement, Risques et Santé 2(1).
Bois, F. Y. and D. Maszle (1997). "MCSim: a simulation program." Journal of Statistical Software 2(9)
Gerlowski, L. E. and R. K. Jain (1983). "Physiologically based pharmacokinetic modeling: principles and applications." Journal of Pharmaceutical Sciences 72: 1103-1127.
Van der Molen, G. W., S. A. L. M. Kooijman and W. Slob (1996). "A generic toxicokinetic model for persistent lipophilic compounds in humans: an application to TCDD." Fundamental and Applied Toxicology 31: 83-94.
Van der Molen GW, Kooijman BALM, Wittsiepe J, et al. Estimation of dioxin and furan elimination rates with a pharmacokinetic model. JOURNAL OF EXPOSURE ANALYSIS AND ENVIRONMENTAL EPIDEMIOLOGY 10 (6): 579-585 Part 1 NOV-DEC 2000.
Van der Molen GW, Kooijman SALM, Michalek JE, et al. The estimation of elimination rates of persistent compounds: A re-analysis of 2,3,7,8-tetrachlorodibenzo-p-dioxin levels in Vietnam veterans CHEMOSPHERE 37 (9-12): 1833-1844 OCT-NOV 1998.

[1] Aylward LL et al. (2005a) Concentration-dependent TCDD eliminationi kinetics in humans: toxicokinetic modeling for moderately to highly exposed adults from Seveso, Italy, and Vienna, Austria, and impact on dose estimates for the NIOSH cohort. Journal of Exposure Analysis and Environmental Epidemiology 15: 51-65.

[2] Aylward LL et al. (200b) Exposure reconstruction for the TCDD-exposed NIOSH cohort using a concentration- and age-dependent model of elimination. Risk Analysis 25: 4: 945-956.

[3] ttp://toxi.ineris.fr/activites/toxicologie_quantitative/toxicocinetique/modeles/dioxine/sub_dioxine.php

[molen-4] 4.0 ^4.1 ^4.2 ^4.3 ^4.4 Van der Molen, G. W., S. A. L. M. Kooijman and W. Slob (1996). "A generic toxicokinetic model for persistent lipophilic compounds in humans: an application to TCDD." Fundamental and Applied Toxicology 31: 83-94.

[5] Gerlowski, L. E. and R. K. Jain (1983). "Physiologically based pharmacokinetic modeling: principles and applications." Journal of Pharmaceutical Sciences 72: 1103-1127.

[6] Bois, F. Y. and D. Maszle (1997). "MCSim: a simulation program." Journal of Statistical Software 2(9): [1].

[7] Bois, F. Y. (2003). "Modélisation toxicocinétique de la concentration sanguine de 2,3,7,8-tetrachloro-p-dioxine après ingestion chez la femme." Environnement, Risques et Santé 2(1). [Toxicokinetic modelling of 2,3,7,8-tétrachloro-p-dioxin blood concentration after ingestion by women. Environnement, Risque et Santé, (2003) 2:45-53. ]

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Human PBPK model for dioxin

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Tuomisto et al model

Dioxin kinetics with Bayes

Aylward et al model

TOXI/INERIS model

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Tuomisto et al

TOXI/INERIS

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