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.
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 
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.
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 
Aylward et al model
TOXI/INERIS model
Dioxin PBPK model 

The results are total amounts (ng) of dioxin, except blood concentration (ng/l). 
The input data used for this variable:

Rationale
Tuomisto et al
The generic onecompartment model says that at a steady state, input and output mass rates of the compound are equal:
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 D = C k V,}
where C is concentration and thus equals m / V, k is elimination constant, V is distribution volume, and m is the mass of compound in the body. Note that this assumes that V is constant, and adjustments must be made if it changes (e.g. body weight increases).
This model is based on onecompartment kinetics assuming that dioxin in evenly distributed in fat in the body, and assuming a dioxin intake trend with a constant relative change r in time. t_{i} is a timepoint (typically in history) and t_{n} means "now". The concentration of dioxin in the body now because of a dose D_{i} at timepoint t_{i} (c_{n,i}) is calculated as follows, assuming that k is the elimination constant for dioxin, D_{n} is dose now, and the dose during time changes at a constant rate r (we assume that dose D was repeating at every timepoint between i and now, just gradually changing in intensity). We also assume that the distribution volume (in practice, amount of fat in the body) is constant. For impact of body fat, see below. All changes (elimination and intake) are relative and thus follow exponential function.
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_{n,i} = \frac{D_i}{V_i} e^{k(t_n  t_i)} = \frac{D_n e^{r(t_n  t_i)}}{V_n} e^{k(t_n  t_i)} = \frac{D_n}{V_n} e^{(r+k)(t_n  t_i)}.}
The total sum dioxin concentration now C_{n} = Σ_{i} C_{n,i} from all doses during the whole exposure period (t = t_{n}  t_{s} is the duration, i.e. time between the starting point of exposure t_{s} and now; and T = t_{n}  t_{i}) 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 C_n = \int_0^t \frac{D_n}{V_n} e^{(r+k) T}dT = \frac{D_n}{V_n} (\frac{e^{(r+k) t}}{(r+k)}  \frac{e^{(r+k) 0}}{(r+k)}) = \frac{D_n (1  e^{(r+k) t})}{V_n(r+k)}.}
Current intake can be solved from here, given current dioxin concentration:
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 D_n = \frac{C_n V_n(r+k)}{1  e^{(r+k) t}}.}
Current dioxin concentration can also be given in terms of intake at start time t_{s} rather than intake now, as D_{n} = D_{s} e^{r t}:
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_n = \frac{D_n (1  e^{(r+k) t})}{V_n(r+k)} = \frac{D_s e^{r t} (1  e^{(r+k) t})}{V_n(r+k)} = \frac{D_s (e^{r t}  e^{k t})}{V_n(r+k)}.}
Impact of body fat
The elimination of dioxins occur almost exclusively in the liver, and therefore liver concentration is the physiological determinant of this firstorder kinetic elimination. Dioxin liver concentration correlates very well with fasted blood concentration and fat tissue concentration but it is not obvious how it correlates with body burden if fat weight varies. So, we must analyse what happens if the amount of body fat changes during the followup period.
If we think about fat weight change as a process where the distribution volume increases in such a way that (otherwise unchanged) concentration decreases exponentially, then we can describe this dilution as if it was another elimination process with elimination constant a. Elimination processes are mathematically straightforward, as the total elimination constant is simply the sum of all elimination constants, i.e. k + a in this case. Note! Although the elimination rate is easy to calculate, the cumulative elimination is not. This is because rate k tells how much goes out of the body, but a tells how much goes back to the fat tissue, which still needs to be eliminated at some point.
Onecompartment model assumes that all dioxin that accumulates in fat tissue is still readily available and in constant balance with blood (and thus liver) concentration, and therefore elimination is proportional to blood and fat concentrations, which are equal (on fat basis). In twocompartment model they are typically different.
If we know the fat weight at time i and now, we can estimate:
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_n = V_i e^{a t}}
where V_{i} and V_{n} are distribution volumes at i and now, respectively. By solving we get
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 ln\frac{V_n}{V_i} = at \Leftrightarrow a = \frac{ln \frac{V_n}{V_i}}{t}.}
For a previous attempt to calculate this using exponential integral function, see[2].
Clearance and changes in distribution volume
Previously, we have assumed that all elimination is proportional to dioxin concentration in the body and distribution volume is constant. However, when compound halflife is several years, it is reasonable to assess situations where distribution volume (body fat in this case) changes. We need a concept of clearance P, which is the volume of blood (mass of fat in blood in this case) that goes to liver and is purified from dioxins per unit of time. In onecompartment model, when V is constant, we have
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 k = \frac{\delta C}{C \delta t} = \frac{\delta m/V}{m/V \delta t} = \frac{\delta m}{m \delta t},}
and
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 P = k V = \frac{\delta m/V}{m/V \delta t} V = \frac{V \delta m}{m \delta t} = \frac{\delta m}{C \delta t}.}
It is reasonable to assume that if a person gains or loses fat, the blood flow and elimination capacity in liver are more constant than dioxin concentration in blood or distribution volume (body fat). Therefore, For kinetic modelling, we could assume that when distribution volume changes, cleareance P, rather than elimination rate k, remains constant. Therefore, we want to adjust our equation so that it only contains parameters that are constant or change in a known way.
Here we assume that if fat mass changes, it happens suddenly at a particular point in time and stays constant otherwise, and the relationship between distribution volumes at t_{s} and t_{n} still follow equation V_{n} = V_{s} e^{a t}. We assume the distribution volume to be constant because if we would let it change gradually, the integral of the equation would not be solvable (according to Wolframalpha [3]).
First, we assume that the sudden change in the distribution volume occurs on the very last day. Then, the V in P = k V equals V_{s}. If a person gains weight, this assumption overestimates the elimination and underestimates the concentration now.
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_n = \frac{D_n (1  e^{(r+k)t})}{V_n (r + k)} = \frac{D_n (1  e^{(r+P / V_s)t})}{V_n (r + P / V_s)} = \frac{D_n (1  e^{(r+P / (V_n e^{a t}))t})}{V_n (r + P / (V_n e^{a t}))} = \frac{D_n (1  e^{(r+\frac{P}{V_n} e^{a t})t})}{V_n r + P e^{a t}}.}
Then, if we instead assume that the sudden change occurs on the first day, V equals V_{n} and we get a simpler equation
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_n = \frac{D_n (1  e^{(r+\frac{P}{V_n})t})}{V_n r + P}.}
In this case weight gain leads to underestimation of elimination and overestimation of concentration. The truth should be somewhere between the two estimates. Because it is not worth using the precise integral, we should use both of the above equations to see whether they make any difference (of course, if the fat mass does not change, they are equal). Clearly, it is straightforward to build a model to solve this numerically by calculating the concentration stepbystep for each year of the followup period. Then, an arbitrary (e.g., databased) function for distribution volume can be used.
Impact of variables on concentration
⇤#: . These equations refer to a previous version and should be checked. Jouni (talk) 14:07, 9 June 2016 (UTC) (type: truth; paradigms: science: attack)
What is the impact of different variables on the concentration of dioxins? For simplicity, let's assign α = D_{n}/V_{n} and β = a  r  k, and t = t_{i}. Then we get
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_{n,i} = \frac{\alpha (e^{\beta t}  1)}{\beta}, \beta \neq 0, t > 0}
We can calculate derivatives of C_{n,i} in respect of first α and then β and we get
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}{\delta \alpha}C_{n,i} = \frac{\delta}{\delta \alpha}\frac{\alpha (e^{\beta t}  1)}{\beta} = \frac{e^{\beta t}  1}{\beta}.}
Because t is always positive, the numerator is negative iff β is negative, and then the whole formula is positive because then also the denominator is negative. In other words, the derivative is always positive and the concentration always increases when α increases.
For β we get
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}{\delta \beta}C_{n,i} = \frac{\delta}{\delta \beta}\frac{\alpha (e^{\beta t}  1)}{\beta} = \frac{e^{\beta t}(\beta t  1) + 1}{\beta^2}.}
In this case the denominator is always nonnegative and does not affect the sign of the derivative. When we test the numerator with several values of t, we can see that it always gets positive values except that it is zero when β is zero (but then the whole equation is not defined). In other words, when β increases, C_{n,i} always increases (or stays the same).
It is important to notice, that in both cases the derivatives are not definded when β is zero, but the limit values from both left and right are the same, so there are no sudden jumps in the function. WolframAlpha was used to produce the derivatives and plot then as function of β and t.
In conclusion, we can say that if we change one parameter and all else remains the same, increasing D_{n} and a lead to increased concentrations now, while decreasing V_{n}, r, and k lead to increased concentrations now. How large an impact will occur depends on the values of other parameters.
More complex situations
⇤#: . This is based on an old version and should be checked. Jouni (talk) 07:56, 10 June 2016 (UTC) (type: truth; paradigms: science: attack)
There may also be a single additional dioxin bolus at timepoint A, and this is denoted d_{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,i} = \frac{D_n}{rk} (e^{(rk) t_i}1) + b_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,i} = (\frac{D_n}{rk} (e^{(rk) t_i}1) + b_A e^{k t_A}) (1  (1e^{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. ⇤#: . There is an error in the equation with breast feeding? Jouni (talk) 11:43, 22 March 2016 (UTC) (type: truth; paradigms: science: attack)
From this, we can solve D_{n}:
 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 D_n = \frac{(rk) (M_{n,i}  b_A e^{k t_A} (1  (1e^{k_B t_B})(e^{k t_S})))}{(e^{(rk) t_i}1) (1  (1e^{k_B t_B})(e^{k t_S}))},}
which can then be used as an independent variable in food intake regression models. This simplifies to
 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 D_n = \frac{(rk) M_{n,i}}{(e^{(rk) t_i}1) (1  (1e^{k_B t_B})(e^{k t_S}))},}
if there is no bolus, and even further
 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 D_n = \frac{(rk) M_i}{e^{(rk) t_i}1},}
if there is no breast feeding.
If the intake is constant and there is no breast feeding, the previous equation solving M_{i} 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,i} = \frac{D_n}{k} (1  e^{k t_i}) + b_A e^{k t_A},}
where D_{n}/k is the steady state amount of dioxin in the body at constant intake, (1  e^{k ti}) is the relative deviation from the steady state, and b_{A} e^{k tA} is the burden from the additional dioxin bolus at A.
Bayesian hierarchical model
Assumptions
 WHOTEQ 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 WHOTEQ
 high intake TCDD 32, 12378PCDD 32, 123678HxCDD 8, 1234678HpCDD 0,1, OCDD 0,3, TCDF 12,6, 23478PeCDF 76 pg/d WHOTEQ
 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 (d1) 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. 2009:117:417425
TOXI/INERIS
 This model description has been taken from an equivalent page of the IEHIASproject. 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, 2,3,7,8tétrachloropdioxin (TCDD), PBPK
Scope of the mode
Physiologically Based Toxicokinetic (PBTK) model for Dioxine. The presented PBTK model allows to simulate ingestion exposures to 2,3,7,8tétrachloropdioxin (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 timevarying 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 shortterm and longterm 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 longterm.
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 min1, 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 2,3,7,8tétrachloropdioxin (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 timevarying 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 shortterm and longterm 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 2,3,7,8tétrachloropdioxin (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, wellperfused tissues and poorlyperfused tissues) as a function of time.
Description of processes modelled and of technical details
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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.
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 (min1).
 (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 halflife 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 longterm.
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 (t11.33)))^{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{t10}{15}, \qquad\text{if }10 \le t \le 15\\0.5+4.5+8+17\times\frac{t15}{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 (t6.5582)))^{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 (t12.478)))^{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 min1, 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).
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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 timevarying 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 shortterm and longterm 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: [4] and [5]
See also
References
 ↑ Aylward LL et al. (2005a) Concentrationdependent 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: 5165.
 ↑ Aylward LL et al. (200b) Exposure reconstruction for the TCDDexposed NIOSH cohort using a concentration and agedependent model of elimination. Risk Analysis 25: 4: 945956.
 ↑ 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: 8394.
 ↑ Gerlowski, L. E. and R. K. Jain (1983). "Physiologically based pharmacokinetic modeling: principles and applications." Journal of Pharmaceutical Sciences 72: 11031127.
 ↑ 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 2,3,7,8tetrachloropdioxine après ingestion chez la femme." Environnement, Risques et Santé 2(1). [Toxicokinetic modelling of 2,3,7,8tétrachloropdioxin blood concentration after ingestion by women. Environnement, Risque et Santé, (2003) 2:4553. ]
 Bois, F. Y. (2003). "Modélisation toxicocinétique de la concentration sanguine de 2,3,7,8tetrachloropdioxine 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: 11031127.
 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: 8394.
 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): 579585 Part 1 NOVDEC 2000.
 Van der Molen GW, Kooijman SALM, Michalek JE, et al. The estimation of elimination rates of persistent compounds: A reanalysis of 2,3,7,8tetrachlorodibenzopdioxin levels in Vietnam veterans CHEMOSPHERE 37 (912): 18331844 OCTNOV 1998.