P.1 

OAR Box 1112 

Prepped by Ryan Dugan 

Document Number: 

29) II-C-1 g 

Docket Number: 

A-83-22 

Printed 4/16/2008 9:12:07AM Sheet 29 of 97 
DESCRIPTION OF LEAD PHYSIOLOGICAL MODEL 

W.F. Biller 

August 1983 

P.2 

A'*-* 

i '• . ^ 

77 - C <1 
P.3 

DESCRIPTION OF LEAD PHYSIOLOGICAL MODEL 

The following is a description of the physiological model 

which will be used with the NAAQS Exposure Model (NEM) to estimate 

the body burden of lead in populations exposed to lead compounds. 

NEM applied to estimating lead exposures will track the 

daily exposure to lead of cohorts from the time of birth to the 

seventh birthday. This exposure will include airborne lead 

breathed into the lungs and lead coming into the mouth from 

ingestion of food, drink and other materials (e.g., soil, dust). 

The details of the NEM model applied to lead are given in a 

separate report (1). Given this daily uptake of lead by a cohort, 

and certain physiological parameters characterizing the cohort it 

will be possible to estimate for every day in the first seven years 

of the cohorts life the amount of lead in various parts of the 

body. The physiological model for performing these computations is 

based on the compartmental analysis model developed by Rabinowitz 

and others (2,3). The compartmental-type model is well suited to 

use with NEM. 

The compartmental analysis approach divides the body into 

a number of well defined parts called compartments. It is assumed 

that lead brought into the body either through the lungs or the 

mouth is dynamically distributed among these compartments and 

ultimately, given sufficient time, excreted from the body. Figure 

1 shows the compartments which are included in the physiological 

model. It is seen that there are four compartments: 

1. Blood 

2. Bone 

3. Soft tissue 

4. Teeth 

Following the assumptions of Rabinowitz, it is assumed the lead 

coming into the lungs and mouth which is eventually taken up by the 
P.4 

above compartments first comes into blood. The arrows in the 

diagram are meant to indicate that there is a dynamic and continual 

exchange of lead between the blood compartment and each of the 

other three compartments. There is no direct exchange between any 

of the other compartments. All lead entering the bone must 

come to it from the blood and all lead leaving the bone must first 

pass into the blood. The same is true for the teeth. In the case 

of the soft tissue all lead entering the soft tissue enters from 

the blood compartment. However, although most of the lead leaving 

the soft tissue does so by passing into the blood some also is 

excreted from the body in sweat or passed into the fingernails and 

hair. Lead is also removed from the blood by the kidneys and 

secreted in the urine. 

As suggested in Figure 1, not all the lead breathed into the 

lungs enters the blood. Some lead is not captured on the walls of 

the lung and therefore is exhaled. More important, removal 

mechanisms operative in the lungs bring lead into the throat where 

it may be expectorated or swallowed. In the latter case it has 

another chance to be absorbed into the blood during passage through 

the digestive tract. Furthermore, only about one tenth of the 

ingested lead passes into the blood as it moves through the 

digestive tract. Since NEM will provide the model with the daily 

quantity of lead taken into the lungs and mouth, the physiological 

model will adjust these inputs to account for the loss of lead from 

the lungs, the rerouting of some lead from the lungs to the 

digestive tract, and the incomplete absorption in the digestive 

tract. 

While entrance of lead into the blood compartment through the 

routes mentioned above is quite rapid, the exchange of lead between 

te compartments and its excretion once in one of the compartments 

is a relatively slow process. The overall half-life of lead in the 

blood is about 30 days, in the soft tissue it is about 40 days, 

while in the bone it appears to be in excess of 5 0 years (2). 

It is assumed that the processes governing the exchange of 

lead between the compartments and excretion are linear, that is, 

they can be described by a system of linear differential equations. 
P.5 

The rate at which lead moves from Compartment #1 into another 

compartment is proportional to the lead content of Compartment #1. 

The proportionality constant is dependent on the direction of the 

flow, the compartment from which the lead is exiting, and the 

department it is entering. The constant for the flow from 

compartment i to compartment j is defined as k^ ^. The studies 

cited above support a linear model. The compartmental system 

illustrated in Figure 1 is described by the following equations: 

(la) dxx/dt = A - k1#1 xx + k2fl x2 + k3fl x3 + k4fl x4 

(lb) dX2/dt = + k1 2*i - k2 2X2 + 0 + 0 

(lc) dX3/dt = + k3 -JX-L + 0 - k3 3X3 + 0 

(Id) dX4/dt **** + k4fl Xx + 0 + 0 - k4f4 X4 

where X^ is the total quantity of lead in compartment i, A is the 

total quantity of lead entering the body per unit time, and k^^ 

is the sum of all the proportionality constants representing removal 

processes for compartment i, that is: 

(2a) klfl - klfQ + kl f 2 + 1*1,3 + kl r 4 

(2b) k2^2 = k2^0 + k2 # 1 

(3c) k3 f 3 = k3 / 1 

(4d) k4 f 4 = k4 f l 

Since lead uptake is constantly varying, difference equations 

provide a convenient approximate solution to the system of 

equations (la) to (Id). We can write: 

(3a) X x ( t i+1 ) = A(ti) + X1(ti) - k-^iX^ti ) + k 2fl X 2 (ti ) + 

k

3,l x3<i >

 + k 4,lX 4 ( i ) 

(3b) X 2 ( t i+1 ) = X 2 ( t i ) + ki^Xl^i ) " k 2,2 X2*.ti) 

(3c) X 3 ( t i + 1 ) = X 3 ( t i ) + k 3#1 X 1 ( t i ) - k 3/3 X 3 ( t i ) 

(3d) X 4 ( t i+1 ) = X 4 (ti ) + k 4fl X 1 ( t i ) - k 4f4 X 4 ( t i ) 

As written the time difference t^+^-t^ is equal to one. 

Notice that it is assumed that A varies with time. Thus, if time 
P.6 

is divided into intervals which are small compared to the time 

constants in the system, and the X^ are all known at the start of 

an interval, Equations (3a-3d) can be used to calculate the X^ at 

the end of the interval. The smaller the time interval compared to 

the time constants k the more accurate the approximation. Given 

that the smallest half-life is about 30 days, a one day time 

interval has been estimated to yield a reasonably accurate 

approximation. Furthermore, if A(t) is known, the Xj(t^+^% 

concentrations can be substituted into the right side of the 

equations to yield the concentrations at t1+ . This process can be 

carried out over an entire time domain for which A(t) is known. 

The errors in this approximation do not accumulate over time. 

An alternative approach would be to assume that A(t) is 

accurately approximated by the average uptake rate per day and 

use the exact solutions to Equations (1). This would result in 

exponential terms in place of the simple linear terms in Equations 

(3). Given the large number of individual computations the model 

will make in following a large number of cohorts over seven years, 

the added calculations time with the exact solutions is probably 

not worth the gain in accuracy. 

The exposure model will allow the computation of A(t) for 

each day in the life of each cohort up to seven years. Each cohort 

will have its own set of time constants k1'^. Furthermore, these 

time constants will change as the cohort ages. Following the 

premises of the exposure model, it will be assumed that the time 

constants for a cohort are constant over three month intervals. 

If the computation is to include the unborn fetus, uptake will 

be from the mothers blood and changes in the time constants will 

have to be at a very frequent basis in the early stages of fetal 

development. 

If the computation starts at birth, a set ofinitial conditions 

for the quantity of lead in each of the body compartments will 

have to be assumed at the time of birth. 

William F. Biller 

8/29/83 
P.7 

REFERENCES 

1. W. F. Biller, "Description of Lead Exposure Model", Document 

submitted to Standards and Strategies Division of the Office 

of Air Quality Planning and Standards, Environmental 

Protection Agency, Research Triangle Park, NC, August 29,1983. 

2. M. B. Rabinowitz, G. W. Wetherill, and J. D. Kopple, "Kinetic 

Analysis of Lead Metabolism in Healthy Humans", Journal of 

Clinical Investigation

 58 , pp. 260-270, August 1976. 

3. S. R. Bernard, "Dosimetric Data and Metabolic Model for Lead", 

Health Physics

 32, pp. 44-45, 1977.