Chapter 19
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Comparison of Symbolic, Numerical Area Under the Concentration Curves of Small Linear Physiologically Based Pharmacokinetic Models Blaine L. Hagstrom, Robert N. Brown, and Jerry N. Blancato Characterization Research Division, National Exposure Research Laboratory, U.S. Environmental Protection Agency, 944 East Harmon, Las Vegas, NV 89193-3478 Using the area under the concentration curve (AUC) dose metric common in physiologically based pharmacokinetic (PBPK) models for human risk assessment, this paper compares recently developed exact symbolic solutions for linear PBPK models with conventional approximate numerical solutions. Comparisons are given for both 4 and 8-compartment PBPK models (P4 and P8). Relative error comparisons are presented for all exposure route, target organ, combinations for P4 and P8. AUC formulas are decomposed into the sum of independent toxic pathways, each pathway the product of 3 physiologically meaningful symbolic factors. The effect of extreme parameter perturbations upon relative error is explored and user options for recovering accuracy are discussed. The complementary role of symbolic and numerical solutions is emphasized. To improve the understanding, validation, uncertainty analysis and general scientific basis of its risk assessment processes in exposure analysis, dose-response prediction, and risk characterization, the U.S. Environmental Protection Agency (EPA) is investigating new diagnostic tools for the analysis of physiologically based pharmacokinetic (PBPK) models describing the exposure and disposition of environmental toxicants in humans and animals. Such tools also help formulate cost-effective regulatoryriskprevention and management strategies. PBPK models examine toxicant dose at all physiological scales, including systemic exposure, target tissue dose and sub-cellular response. PBPK models help refine the qualitative formulation of biological processes, their quantitative mathematical computation, and their sensitivity to model, parameter and data uncertainties. PBPK models are described by mixed systems of linear and nonlinear ordinary differential equations (ODEs) which often reduce to purely linear systems at low concentrations typical at most human and some test animal exposures. Both toxicant concentrations and integrated concentrations are commonly computed using general purpose stiff numerical ODE algorithms such as Gear's [3]. In previous work with a 4-compartment PBPK prototype (P4) [1], symbolic analytic dose formulas were developed for the area under the concentration curve (AUC) dose metric common in quantitative risk assessment (QRA). 0097-6156/96/0643-0256$15.00A) © 1996 American Chemical Society In Biomarkers for Agrochemicals and Toxic Substances; Blancato, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
Downloaded by NORTH CAROLINA STATE UNIV on September 23, 2012 | http://pubs.acs.org Publication Date: September 27, 1996 | doi: 10.1021/bk-1996-0643.ch019
19. HAGSTROM ETAL.
Symbolic, Numerical AUCs and PBPK Models
257
Concurrently in these Proceedings, a more intuitive terminology, consistent physiological formulation, and efficient symbolic AUC solution was presented for P4, and extended to selected inputs and outputs of an 8-compartment PBPK model (P8) [R.N. Brown; 'Symbolic AUC Solutions of Small Linear PBPK Models Useful in Risk Assessment and Experimental Design']. Those AUCs were observed to be decomposable into the product of three physiologically meaningful factors: a crude AUC, a fractional conductance, and a circulation multiplier. The main purpose here is to compare, for all exposure route, target organ combinations for P4 and P8, the numerical AUC accuracy of approximate Gear-type ODE solvers with exact symbolic formulas. We further explore the physiological structure of symbolic AUCs. The complementary roles of symbolic and numerical solutions for risk assessment and management applications are emphasized. We also explore, via extreme fat partition coefficient perturbations, numerical solver breakdown and error-control options for accuracy recovery. All ODE solver and symbolic evaluations were performed using SimuSolv software [2], but P4 symbolic results were also verified by hand evaluation of symbolic formulas. Review of P4 Model and Symbolic AUCs The P4 prototype consisted of (i) a main compartment (A = composite arterialvenous blood); (ii) a slowly perfused strong sink compartment (F = fat); (iii) a slowly perfused weak sink compartment (M = muscle); and (iv) a large flow systemic clearance compartment (L = liver). The mass balance equations were given previously in low concentration linear limit form (inputs suppressed) by
VA
VA \
'F
A
r> -
,
°
L
M
_A'
1
L
~ V
L
_
A!
i
n
R
HAF
Q^CA
~ V-RLT
HAM
HAL
* QFACA
~V
L
VA \
KA
J
1 (QAFC?
1 QMACA
1
/
1
\
(QAMCM
,
(QALC
,
+
v
^
+
\
N
iffi
L
[ l ^
T /
C
L
T ^ T ^ £ j
\
M
( 1 )
where clearance terms (in vol./time units) are given by QM\ by Vikd where is a kinetic transfer rate for volume VJ; or by V^/K when the nonlinear perturbation term 1 -I—• 1, where is the amount form maximum metabolic elimination velocity (in mass/time units) and K is the Michaelis constant (in mass/vol. units). Rij is the unitless partition coefficient describing the ratio of toxicant concentration in compartment j relative to i. In matrix form, the mass balance equations, for input vector i , are given by mi
mi
C = -V-\CL)C
+ i,
X
(2)
where C = V~ A is the concentration derivative vector (in mass/volume-time units); A' the amount vector; V " the diagonal matrix of inverse volumes; and —CL the negative system clearance matrix given by 1
In Biomarkers for Agrochemicals and Toxic Substances; Blancato, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
258
BIOMARKERS FOR AGROCHEMICALS AND TOXIC SUBSTANCES QAT.
AL
R
o -CL
=
9** A
0
MA
R
0
/ —CLtA CLFA C L
CLAM
CLAL
\
—CLtF
0 0 (3) 0 —CLtM 0 CLLA 0 0 -CLu. / with off-diagonal elements (CLij) representing local circulatory clearance to compartment i from j and diagonal elements CLu (= CLu) denoting total clearances from t. Previously CL was rewritten in t-normalized matrix form as M
Downloaded by NORTH CAROLINA STATE UNIV on September 23, 2012 | http://pubs.acs.org Publication Date: September 27, 1996 | doi: 10.1021/bk-1996-0643.ch019
CLAF
v
A
y
CL = StCU -*AF
1A
IF
—TtMA —TtLA
0 0
—*AM -*AL \ 0 0 0 1M 0 u y V
0 0 0 0 0 CLtF 0 0 CLtM 0 0 CLtL
0 0 o
(4)
th
where S is CL with 1 column divided by CLu so that diagonal entries are unity (TTtt = It) and so that off-diagonal entries represent the fractional clearance to compartment i from j (n^ = CLij/CLtj = kij/k j). S is the system's total- or t-normalized unitless structural interconnection matrix, which emphasizes the importance offractionalflowsbetween compartments in model formulation, understanding and solution. Figure 1 shows the P4 system graph useful in summarizing symbolic AUC solutions. AUC, a useful internal dose refinement of external exposure for studying many intermediate or long term health effects, especially those independent of shortterm exposure variations, was given previously as t
t
AUC
= j H C(r)dr =
=
CL~ AQ X
t
CL^S^Ao,
(5)
generalizing the classical 1-compartment formula AUC = AQ/CL [4]. Using matrix row-column determinant expansions, the finite, nonnegative, t-normalized single-input, single-output (SISO) components for an I compartment output from a 3 compartment input were given [1] by B
th
th
(6) if \S \ = 1 - TCAF^FA - TTAMTTMA - TtALitLA > 0 (i.e., if S invertible). For all P4 and selected P8 inputs and outputs, this reduced to the more intuitive t
t
\S -Ki...j\ AJO t
\S \ T
Audi =
(7)
CL
U
52*^
In Biomarkers for Agrochemicals and Toxic Substances; Blancato, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
19. HAGSTROM ET AL.
Symbolic, Numerical AUCs and PBPK Models
259
where overall AUC reduced to the sum, over all possible toxic pathways from j to i , of 3 physiologically meaningful factors: (i) where 7^...;, denoted the fractional conductance (i.e., product of individual fractional clearances) along a particular pathway; (ii) where the determinant ratio \S — / \St\ denoted the circulation multiplier, describing the effective average number of times the parent toxicant reentered the input compartment j for repeat conductance down the toxic pathway to target organ i\ and (iii) where AJQ/CLU denoted the crude AUC, as if all input went directly to the target organ. Note that \S — ir%...j\ = \S — (t...i)| represented the residual subdeterminant after subtracting from S the (i...j) rows and columns making up the toxic pathway. t
t
t
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t
P4 Symbolic, Numerical A U C Comparisons For comparing analytic and numerical AUCs, nondefault independent parameters for P4 are given in Table I (default partition coefficients = 1, default systemic clearances = 0). Dependent parameters for symbolic AUC construction are given in Table II. Numerical, symbolic AUC computations are given in Table III in the 'Approx.' and 'Exact' columns for all 4 SISO AUC outputs from an arterial bolus input (Aio). Table truncation anomalies, involving apparent repeating decimals, are corrected using overbar notation in examples below. Interpretation is simplified by giving the relative error between approximate and exact AUCs, along with numerical evaluation of the 3 factors making up each AUC subpathway. Bolus muscle, fat and liver input results are given in Tables IV, V and VI.
QFA QLA
= 25 = 100
Table I: P4 nondefault independent parameters. V> = 13 RAL = 10 RAF QMA = 275 Vk = 5 VL = 2 V = 50 RAM = 10 CL QM = QiA
100
= =
EL
M
45
Table II: P4 dependent parameters. = 0.1818 *LA = 0.2500 TTAF = 1.0000 TT = 0.0625 1T = 1.0000 TT A = 0.6875 CL =25.00 CL i=275.00 CL =100.00 CLvu^O.25 CL,t =27.50 ClM =10.00 CLt =27.50 CLtF=0.25 01^=400.00 CLt =55.00 TTAL
FA
AM
M
tA
Fj4
M
M
My
t
L
Table III: P4 A U C M for bolus Arterial input. RelErr.= C o n d ^ T T ^ , M u l t . - l S t a ^ , C r u d e = ^ , A = 0.1. Exact RelErr. i Approx. Cond. M u l t . A 0.00122222 0.00122222 5.7149E-7 1.000000 4.88889 P 0.12222200 0.12222200 3.6576E-7 0.062500 4.88889 H 0.01222220 0.01222220 2.2860E-7 0.687500 4.88889 L 0.00222222 0.00222222 5.2387E-7 0.250000 4.88889 A0
Crude 0.00025 0.40000 0.00364 0.00182
For all P4 tables, the salient feature is the small relative error of the approximate AUCs. Looking closer at AUC structure, the baseline example of an arterial
In Biomarkers for Agrochemicals and Toxic Substances; Blancato, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
260
BIOMARKERS FOR AGROCHEMICALS AND TOXIC SUBSTANCES
AUC from an arterial input is given from (6, 7) and Table III as
with fractional conductance TTAA ~ 1 since input is to target organ; circulation multiplier 1/ \St\ = 1/.204545 = 4.89, where residual numerator subdeterminant \S — A| = 1 since (S — A) is diagonal; and crude AUC small (AAo/CL A = .00025). For a weak sink (moderate partitioning), length-1 toxic pathway, Downloaded by NORTH CAROLINA STATE UNIV on September 23, 2012 | http://pubs.acs.org Publication Date: September 27, 1996 | doi: 10.1021/bk-1996-0643.ch019
t
T
t
with crude AUC 15 times larger than baseline (at .00364 vs. .00025), conductance 1/3 smaller (at .6875 vs. 1), and AUC 10 times higher (at .01222 vs. .001222). For a strong sink example, the fat AUC from a bolus arterial input is given by wc
F A
_
.
- .na.
(io,
with crude AUC about 1000 times higher (at .4) and conductance about 10 times smaller (at TTAF = .0625), to yield AUC 100 times higher than baseline. For a smaller multiplier, Table IV shows muscle AUC from a bolus muscle input as Aim
Ap
[5 - M\ t
AUC
=
MM
1 - TTAF^FA ~ ^AL^LA
M
CLtM
N
=
N
AQ CUMM
N
-
I
K
Q
-°
159,
with conductance unity (T^MM = 1) and multiplier 4.36 = .89/.204 (
