Appendix C
Detailed Comments on the Integrated Exposure Uptake Biokinetic Implementation Code
The uncertainties and inaccuracies listed below are referenced to sections of the Technical Support Document (TSD) for the integrated exposure uptake biokinetic (IEUBK) model (EPA 1994).

The bone weight (WTBONE, as given by equation B5g (p. A10 of the TSD) is not continuous, because the two equations do not match at 12 months with the given definition for WTBODY. At 12 months, 0.111 × WTBODY = 1.1192265, whereas 0.838 + (0.02 × 12) = 1.078, about 4% lower.

Equation B2b (p. A7 of the TSD) defines TRBCPL as
TPLRBC × (RATBLPL – 0.55/[0.55 + 0.73]).
The text (p. 40 of the TSD) simply states that TRBCPL is the product of TPLRBC and RATBLPL minus a constant, without any explanation why. If TRBCPL is being estimated by the usual assumption that the ratio of TRBCPL and TPLRBC is equal to the steadystate mass ratio (p. 29, paragraph 2 of the TSD), then the “constant” here is not in fact quite a constant, because then:
TRBCPL/TPLRBC = RATBLPL – (VOLPLASM/VOLBLOOD)/ ([VOLPLASM/VOLBLOOD] + [VOLECF/VOLBLOOD])

VOLECF/VOLBLOOD = 0.73 (equation B5d of the TSD), but VOLPLASM/VOLBLOOD is not the constant 0.55 implied in equation

B2b. Although this ratio is fairly constant, it is only as low as 0.55 for ages less than 0.4 month and exceeds 0.6 for all ages between 5 and 84 months (with the parameter values given in equations B5a and B5c). None of this makes any substantial difference, but the discussion on page 29 needs to be amplified to indicate where this “constant” comes from.

On the same matter, to agree with the statement that the ratio of times is equal to the ratio of steadystate masses (p. 29 of the TSD), it should not be the ratio of TRBCPL and TPLRBC that is set to this mass ratio but the ratio of TRBCPL to TPLRBC2, because TPLRBC2 is the actualtime constant.

The definition of TPLRBC2 given in equation B2.5 of the TSD is not physical, since it relates to VOLRBC (t – 1), which presumably is supposed to be the volume of red blood cells at the previous time step, and, of course, the time step of a computer program has nothing to do with the mathematical definition of the problem. It might be a viable approximation in a computer program to use the value in the previous time step, but in the actual computer code, the value in the previous month is used not the value in the previous time step.^{1}

On p. A10 of the TSD, equations B5a, B5b, and B5c define the blood, plasma, and red blood cell volumes, but the required relationship VOLBLOOD = VOLPLASM + VOLRBC does not hold at all times. It is not clear what the difference is supposed to represent. With the values given, this difference turns out to have different signs at different ages, suggesting that the equation just given is supposed to hold (as one would expect, unless there is supposed to be another compartment to hold the other cellular components of blood). This is an example of an unnecessarily introduced approximation that would be trivial to correct.

On p. B7 of the TSD, a definition of HCT0 is given in such a way that numerically it differs from 1 – VOLRBC(0)/VOLBLOOD(0). This is again an unnecessary approximation.

On page A18 of the TSD, the initial conditions are defined. However, the source of these initial conditions is not clear. The statement after equations B7a through B7d that equations B7a through B7d are “numerically equivalent to the following equations” is incorrect. For example, equation B7d could be numerically equivalent to the corresponding equation below only for HCT0 = 1.284, which is physically impossible. B7b could be numerically equivalent to its corresponding equation below only accidentally. Indeed, neither set of equations corresponds to the assump

tions described earlier in the TSD. If some other set of assumptions is being used, then it should be documented how those assumptions lead to the equations of p. A18. In the computer code, both sets of equations are present, and indeed both are executed; but only the second has any effect.

On page A19 of the TSD, equations B7e and B7l contradict the statements made under MCORT(t) on page B9, and MTRAB(t) on page B11. In both cases, it is stated that there is an assumption that the bone (cortical or trabecular) lead concentration/blood lead concentration ratio is equal to the bone (composite) lead concentration/blood lead concentration ratio (so cortical and trabecular bone lead concentration/blood lead concentration ratios should be equal). Equations B73 and B7l give different concentration ratios (78.9 for cortical, 51.2 for trabecular).

Equations B4a through B4d (p. A9) are stated (p. B4 and B5) to come from an analysis of the data of Barry (1981). However, at age 0 they are contradicted by the initialization conditions given in equations B7e through B7l (p. A17), which are said to be based on the same data (p. B9, B10, B11). For kidney, liver, and other tissues, the tissue/blood concentration ratios implied by equations B4a, B4b, and B4d at time 0 are 0.777, 1.1, and 0.931 L/kg, whereas equations B7f, B7g, and B7h give 1.06, 1.30, and 1.60 L/kg, respectively. Here is another internal inconsistency, because equation B4c gives a bone/blood concentration ratio of 6.0 L/kg at t = 0, whereas equations B7e and B7f give separate ratios at t = 0 of 7.89 and 5.12 L/kg for cortical and trabecular bone, respectively.
REFERENCES
Barry, P.S. 1981. Concentrations of lead in the tissues of children. Br. J. Ind. Med. 38(1): 6171.
EPA (U.S. Environmental Protection Agency). 1994. Technical Support Document: Parameters and Equations Used in the Integrated Exposure Uptake Biokinetic (IEUBK) Model for Lead in Children (v 0.99d). EPA/540/R94/040. OSWER 9285.722. PB94963505. Office of Solid Waste and Emergency Response U.S. Environmental Protection Agency, Washington, DC [online]. Available: http://www.epa.gov/superfund/programs/lead/products/tsd.pdf [accessed Jan. 4, 2004].
EPA (U.S. Environmental Protection Agency). 2002. System Requirements and Design For The Integrated Exposure Uptake Biokinetic Model For Lead In Children (IEUBK) Windows Version – 32Bit Version. OSWER 9285.743. Office of Solid Waste and Emergency Response, U.S. Environmental Protection Agency, Washington, DC. May 2002 [online]. Available: http://www.epa.gov/superfund/programs/lead/products.htm [accessed Oct. 11, 2005].