Biomarkers of Human Exposure to Pesticides - American Chemical

Chapter 18

Analytic Solution of a Linear Physiologically Based Pharmacokinetic Model Prototype Useful in Risk Assessment Downloaded by UNIV OF GUELPH LIBRARY on September 9, 2012 | http://pubs.acs.org Publication Date: November 23, 1993 | doi: 10.1021/bk-1992-0542.ch018

Robert N. Brown Exposure Assessment Research Division, Environmental Monitoring Systems Laboratory, U.S. Environmental Protection Agency, P.O. Box 93478, Las Vegas, NV 89193-3478

Explicit analytic solutions of physiologically based pharmacoki netic (PBPK) models yield important qualitative as well as quantita tive information about chemical exposure and risk in the human body beyond that provided by general purpose numerical simulation soft ware for such models and may provide convenient replacements for unwieldly or inaccurate simulation tools in some applications. Biolog ically based risk assessements for chronic toxicity typically utilize inte grated concentration, i.e., area under the concentration curve (AUG), for the internal exposure or dose metric in dose-response risk mod els. Simple matrix based algebraic formulas for target organ AUCs are developed for bolus dose inputs and steady-state infusions for a progression of 1,2,3 and 4 compartment linear PBPK models, paral leling but enhancing analogous algebraic methods for classical com partmental drug models. Interspecies scaling formulas are derived for potential extrapolation of dose from small animals to humans. Com parative dose formulas are given for the relative impact of bolus dose administration by two exposure routes, formulas that may be useful in developing optimal drug efficacy or pesticide useage strategies.

Physiologically based pharmacokinetic ( P B P K ) risk assessment models of toxic compound exposure, internal dose and health effect are typically solved with numerical simulation packages using general purpose ordinary differen tial equation (ODE) discretization methods, such as SimuSolv's Gear algo rithm [5], used by Anderson, et al., for their analysis of methylene chloride toxicity [2]. On the other hand, first integral analytic solutions of general linear matrix O D E systems are well known ([7], [8]) and should provide ac curate time dependent analytic concentration formulas for P B P K models. In this paper, however, our interests focus on longer term quantitative risk

This chapter not subject to U.S. copyright Published 1994 American Chemical Society

In Biomarkers of Human Exposure to Pesticides; Saleh, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1993.

Downloaded by UNIV OF GUELPH LIBRARY on September 9, 2012 | http://pubs.acs.org Publication Date: November 23, 1993 | doi: 10.1021/bk-1992-0542.ch018

302

BIOMARKERS O F H U M A N EXPOSURE T O PESTICIDES

assessment (QRA) applications (e.g., intermediate term drug effects or long latency cancer effects) from chronic exposure to toxic compounds such as agricultural pesticides or fumigants. Thus, our goal will be to adapt classical first and second integral matrix ODE solutions for internal organ dose to second integral area under the concentration curve (AUG) dose measures and to steady-state concentration levels for small linear PBPK models. This will generalize A U C results for nonphysiological classical compartmental models [6]. System based analytic A U C formulas for classical compartmental drug models have ocassionally appeared in the literature, but their systematic exploitation in the P B P K context appears to be insufficiently explored [9] for many risk assessment purposes. This general lack of interest in analytic P B P K solution development may be due to the fact that: (1) P B P K models are considered more complex that classical comparmental models; (2) P B P K model forms have not been well standarardized in matrix terms; (3) the formalistic equivalence between P B P K and classical models is poorly understood; and (4) the second integral analytic solution of P B P K models involves a seemingly unwieldly and computationally inefficient inversion of the system matrix. In addition, the ready availability of general purpose numerical simulation packages for nonanalytic number crunching of integral solutions has tended to discourage the search for, and utilization of, analytic methods for P B P K models, even linear models. However, general purpose numerical software tools yield relatively little conceptual understanding of biological toxicity processes (due to lack of analytic expressions) result in computationally intensive sensitivity analyses (Monte Carlo analyses) and may be computationally inaccurate or inefficient under some conditions (from use offinitedifference approximations or small step sizes). One goal of this paper shall be to address these misperceptions regarding the presumed obsolescence of analytic solutions given the availability of general purpose numerical simulators. It will present, apparently for the first time in a generic P B P K context, simple analytic solutions for a sequence of small dimensional (n< 4) prototypical PBPK models that may be useful in representative QRA applications. Representative applications include (1) identification of sensitive system parameters, (2) inverse reconstruction of exposure inputs from internal system biomarkers or physiological outputs, (3) extrapolation of dose from small animals to humans, and (4) comparison of target organ dose by different routes of toxicant administration. A broader goal is to begin the use of analytic solution methods to motivate more efficient model formulation, validation, manipulation and estimation procedures for PBPK models than is currently available using nonanalytic numerical simulation approaches alone. Furthermore, linear analytic solution methods may motivate development of both more powerful analytic and numerical algorithms for understanding and applying nonlinear PBPK models. Nonlinear Q R A applications might include analytic extrapolation of internal dose from high-to-low exposures and extrapolation of internal dose from


18. BROWN

303

Analytic Solution of a Linear PBPK Model Prototype


continuous to bolus dose exposures - trivial problems in the linear, but not nonlinear, model case. The analysis of P B P K models may be simplified through the use of fac tored matrix formulations. Although not necessary for system solution, fac tored matrices help standarize model form, clarify biological parameter and state variable interrelationships and simplify solution methods. To motivate greater use of factored matrix P B P K models for QRA, it will be helpful to clarify the relation between P B P K and classical compartmental models.

1-Compartment Classical and ρ β ρ κ Model The classical 1-compartment pharmacokinetic model is given by [6]

where A\ is the amount of toxicant in the well-stirred main compartment, k is the elimination rate to the outside and t^(t) is the temporal input function into the main compartment and may be a continuous function or bolus dose. Use of amounts simplifies mass balancing of inputs and outputs, but the biologically convenient concentration form is given by t

ét

= - hv^ =- ^

d

h

c[{t)

hC

H

(2)

where V\ is compartment volume, C% is concentration and i\(t) = %A is concentration input rate in units of mass/volume-time rather than the previous bolus dose input rate, %A (0> which was in units of mass/time. The standard ([6], [7], [8]) first integral solution of (2) is given by X

X

k

s

C (t)= ['- ? 1 2

s=

0

1

RI

V

In -I21 -I31

0

0

R4

1

J

-qu \

122

0

0

l33

0

0

Î44

0

-I41

-?13

0

(25)

0

/

where q = Q / Ç i , 9 i 3 = QzlQuqu = QAIQI such that q + 913 + qu = 1. All vectors and matrices are indexed as in the 3 compartment case, but for 4 dimensions. The inverse structural matrix (S*" )— is given by 12

2

u

1

1

S- =

^ Su Su S\3 S14 S21 S22 S23 S 4

=1234

\

2

\

$31

$32 $33 $34

S41

S42 S43

S44

J


18. BROWN

/

Analytic Solution ofa Linear PBPK Model Prototype

Î22Î33Î44

912I33I44

913I22U4

914I22I33

I21Ç13I44

I21Ç14I33

309

I11I33I44 —Ç14I41133

I21I33I44

—Ç13I31I44

I11I22I44

I31I22I44

I31Ç14I22

—Ç12I21I44 I31912I44

—Ç14I41I22


I11I22I33

U1Î12I3

I41I22I33

—Ç12I21I33

I41Ç13I22

—913I31I22

\s\ where \S\ = I 1 1 Î 2 2 Î 3 3 Î 4 4 - Ç12I21Î33Î44 the A U G components are given by

-

Ç13I31Î22Î44 -

l{" 1

/

D

l

2

Z

l

l

i

(&iQï A o+& Qï A20+$23Q - A o+$ Qï Aioy

R ^

2

Thus,

15|

R

( AUCx \ AUC AUC \ AUC )

Ç14I41Î22Î33.

2

3

3

7i

m

(27)

(îQT Aio+S Q^ A o+S Q3 A3o+S QZ A o) 1

Ώ

I

D

(

1

32

1

2

1

34

33

A

4

R

^

^ (( ^ ^ ^ 1

22

33

(SiiQÂio+SuQÂn+SuQÂw+SHQÂio^

* {

44

^r

l y 4

H2

Λ

\

\

-Ç13I31Î44

1

l2iÎ33Î44Q ~ >lio+ 1

\

Λ

QJ

1

A +I21

-i (l

+

f £ i )

~

/

W°ifk «fcu

I

if k i -

l e

»

*u

λ

J'

(36) where W refers to the body weight of the animal, 6 refers to the scaling constant for the compartment's basal metabolism related "primary" physio logical uptake processes for amounts or volumes, 6—1 refers to the scaling constant for the equivalent concentration uptake processes, and β — I refers to the compartment's concentration scaling constant for secondary eh mi na tion processes, if any. Typically, the primary basal metabolism concentration uptake scale constant is about (6—1) = —.25 and the secondary elimina tion constant is also about (β — 1) = — .25 (if it is also basal metabolism correlated). However, the secondary elimination constant may be closer to (β — 1) = 0 for some subcellular elimination processes such as metabolic clearances processes largely independent of animal body weight and sponta neous thermodynamic elimination processes associated with reactive metabo lite clearance [11]. Indeed, empirical curve fits suggest that 6 = .73 is an appropriate overall interspecies scaling constant for many composite uptake and elimination processes dominated by basal metabolism energy needs [10]. If AUC is independent of body weight (i.e., oc W°), then this is typically referred to as overall basal metabolism scaling. If AUC is proportional to W"- , then this is typically referred to as overall bodyweight scaling. This terminology may be a misleading oversimplification since AUC scaling is proportional to the ratio of composite uptake to composite elimination pro cesses, implying that the overall scaling constant is the difference of uptake and elimination scale constants ((6 — β) = (6 — 1) — (β — 1)). The often cited basal metabolism scaling constant (6 « .75) might be thought of as the observed surface area to volume scaling constant for com monly measured animal shapes rather than that for homogeneous spheres (b=2/3). Commonly estimated scale constants may reflect values greater than 2/3 as the composite result of two separate scaling phenomena: the 25


18.

313


BROWN

scaling of small pudgy animals (e.g., mice, rats, hamsters) to large pudgy animals (e.g., minipigs) and the scaling of pudgy animals to spindle-shaped animals (e.g., monkeys and humans). To extend this scaling formula to larger PBPK models, consider the 4compartment AUC formula (32), which can be reformulated as

ν


x

oc

\ (CjA

1

(hiqii

AUdj

V + w*-» ; 1

v

l u I22

1

+

^ 1 - 1 ;

l u I33

v

1

. . . . . . .

l u I44 / '

w£>

2 2

f u >

3

3

*ν

η

(37) fV J 4

where the term (0 ^ J2k=uj{h — &)) represents a plausible range for the composite sum of three scaling constant differences for the compartments (l,ij), Wa = (l + wPb]Zl) ~ oc W° (i = 1,2, j) if primary elimination, k^ = p^-, dominates secondary elimination k (i.e., \ = (1 + f^) —• 1). How ever, W,, ~ oc W ' where secondary elimination dominates (i.e., l —» 00) and scales as bodyweight (β, — 1 = 0). Significantly, the long denominator term, ( l - ^ ^ - 4 u - J U i - - , appears to have smaller relative efte

u

25

u

feet upon the overall composite scaling constant than the isolated W terms, at least when this long term is bounded away from zero (e.g., when liver flow is large and the liver has dominant secondary metabolism). However, as it approaches zero, the corrective effect of this long term increasingly coun teracts the scaling effect of the separate product factors, ( W ^ W n ^ j j ) This second order effect should be looked at more carefully in specific scal ing situations. The individual (bk — βίζ) terms can theoretically be positive or negative, but positive differences would seem, intuitively, to predominate where secondary elimination is spontaneous (e.g., for reactive metabolites) or metabolic but proportional to cellular volume (i.e., independent of cellular basal metabolism energy needs). To highlight the potential magnitude of effect upon the scaling constant, it may be illustrative to consider a somewhat extreme " double-teamed" case where the target organ and main compartments both eliminate toxicity by bodyweight scaled processes (/?,· — 1 = /?χ — 1 = 0) dominated, say, by sec ondary spontaneous metabolic elimination pathways, but uptake and other elimination processes scale as basal metabolism (/?, — 1 = &i — 1 = 6, — 1 = bj — I = —.25). In this case, the long denominator term (1-...) is near unity and has minimal corrective impact. Then (38) below suggests AUCij oc which would imply that large animals are at far less risk than small animals u

1


1

314

BIOMARKERS OF HUMAN EXPOSURE TO PESTICIDES


m2S

- less than bodyweight scaling (AUCoc W~ ) and much less than basal metabolism scaling (AUCoc W°) would imply. On the other hand, empir ical evidence for human drugs with respect to noncancer health endpoints suggests the use of more conservative (i.e., more protective) human scaling closer to basal metabolism scaling [4]. Furthermore, empirical evidence for cancer endpoints ([1], [4]) suggests scaling constants between bodyweight scaling (b = 1.0) and spherical surface area scaling (b = 2/3). For compari son, overall interspecies scaling constants traditionally used by United States regulatory agencies for cancer risk range from surface area scaling (6 = 2/3) typically used by the Environmental Protection Agency and Consumer Prod uct Safety Commission, to body weight scaling (b = 1.0) used by the Food and Drug Administration and Occupational Safety and Health Administra tion [4]. When input and target compartments coincide, (33, 34) imply that AUG a «

x

W£

oc iy,

(38)

where the term 0 τ=* (6, — /?,·) denotes a plausible range of scale constants. C o m p a r a t i v e Risk ( C R ) Route-to-route Dose Extrapolation.

One

of the most important theoretical and practical uses of analytic PBPK for mulas is to compare the relative impact upon target organ dose from two different routes of toxic compound exposure or bolus dose administration. To clarify the discussion of the comparative dose formulas below, it will be helpful to consider plausible physiological interpretations of each of the four compartments. In particular, it may be helpful to assume that: (1) the first compartment represents the arterial blood compartment, possibly lumped together with the venus blood, kidney, lung or some other rapidly perfused organs, (2) the second compartment represents the lumped liver, gall bladder and perhaps some other rapidly perfused compartments not lumpted with the main blood compartment, (3) the third compartment represents the fat com partment and (4) the fourth compartment represents other lumped slowly perfused organs (e.g., muscle, bone and skin). Depending upon the toxic compound and risk application, appropriate organ lumpings can vary. Representative comparative dose formulas are given below for several cases of potential bolus dose input into two different compartments of the 4compartmental model above. While most of these results may be intuitively understood by researchers utilizing classical compartmental modeling tools, the comparative PBPK formulas below appear to provide more complete quantitative and qualitative tools for internal dose description, biological understanding and risk assessment manipulation. First, for the case of peripheral fat compartment comparative risk (CR) from liver and blood bolus dose toxicant inputs (e.g., by ingestion into liver and inhalation into blood), CR is given by equation (27) as (S- ) Q2 A o l

(s-η

l

32

2

l

Q- A

l0

ï

A


(39)

18. BROWN

Analytic Motion of a Linear PBPK Model Prototype

315

Thus for equal bolus dose inputs, comparitive risk is < 1 (Am = Αχο). In particular, C R = 1 when liver secondary elimination is zero (i.e., 1 = 1), but CR—> 0 as secondary elimination increasingly dominates primary elimina tion (as 1 —• oo). This represents a simple mathematical description of the liver's biologically critical role as a first pass effect in protecting peripheral organs from environmental insults via ingestion relative to insults by inhala tion and shows that only target organ and input compartment parameters effect relative risk in linear models. However, this is not true for absolute A U C where the noncancelling det(S) reflects contributions from all compart ments. Note that comparative risk can be easily computed when bolus inputs are not equal, but some insights may be easier with equal inputs. By symmetry to the above case, 22


22

Also by symmetry, CR^

=

4l

Λ30/Λ10Ϊ33,

which is unity for equal inputs

(A30 ±= Αχό) when fat secondary elimination is zero (Î33 = 1). Second, the liver compartment C R for liver and blood inputs is given by (4)

l

_ AUC22 AUC21

CR

22 21

'

l

(S~ ) Q A (S~% Qï A

=

22

2

In Λ q l \

_

20

l

t

m

l 2

gialai _ quUi\ Λ20 î Î33

Î11Î44M0*

n

2 1

If secondary elimination occurs only in the liver ( 1 φ 1, l u = Î33 = Î44 = 1), then C R = 1 (if A o = Ai ) since the parenthetical term then equals qi . But CR—» 00) than liver. Also, C R = 1 if A o = A30 and liver and fat secondary elimination are both zero. By symmetry, CR^/ \ — ( l 4 4 ^ 3 o ) / ( i 3 3 ^ 4 o ) « Fifth, blood compartment C R for liver and fat inputs is given by 2 2

2

2

_

(A)

RP

u

n

^ -

1

l

AUC _ (S- ) Q A __ quittUAQÂn AUC ~(S-i) Q~ A ~q î î Q- A l2

12

2

20

l

l3

l

30

133^20

î A


f

K

4

« x }

316

BIOMARKERS OF HUMAN EXPOSURE TO PESTICIDES

which surprisingly equals CR^ above. Thus, changing the target organ from the peripheral (muscle) compartment to the main (blood) compart ment does not change comparative risk relative to two other peripheral compartment inputs (i.e., liver and fat)! Also, by symmetry, CR^ = (AUCis/AUCu) = (Î44A30)/(Î33A10),which tends toward unity for equal bolus dose inputs (A3o=A ) since fat and muscle secondary elimination is typically zero. However, CR—• 0 as fat secondary elimination increases faster (as Î 3 3 —• 0 0 ) than muscle, but CR—• 0 0 if the reverse relationship holds. Sixth, the liver compartment C R for liver and fat inputs is given by 43

U


40

(4)

_

CR

22 23

/

AUC22

l

=

l

(S~ ) Q2 A 22

20

l

AUC

23

(S~% Qs A 3

m

=

Î11Î33

Λ _ qnhi

l q { 2 l

t 2

î

N

Î 3 3

_ quUi\

A

î îj

Αχ>'

n

20

(44) which mimics C ^ 2 / 2 1

a D O V e

(40), but increases faster as secondary blood and 1

fat elimination increase together (CR—* qï Î 1 1 Î 3 3 — * 0 0 as I n , I33 — • 0 0 ) . Similar comparisons have been attempted elsewhere for more complex models using general purpose numerical simulation tools [3]. 2

Discussion a n d F u t u r e D i r e c t i o n s Simple 1,2,3 and 4-compartment linear P B P K dose models have been refor mulated in factored matrix O D E form in order to elucidate their essential structure and to simplify their analytic solution to forms potentially more useful for quantitative risk assessment (QRA) and experimental design (ED) applications. Theoretical applications to interspecies scaling of internal or gan A U G dose and to comparative target organ dose or risk from two routes of bolus dose administration have been theoretically explored. Practical ap plications might include development of optimal drug delivery strategies for humans or prioritization of pesticide exposure control strategies for farm workers. Such simple linear exposure models suffer many potential biological weaknesses surrounding the following issues: (1) relative absorption rates for the major routes of administration, (2) organ lumping strategies (i.e., model expansion and collapse), (3) complex compartmental interconnections and feedbacks, (4) well-stirred compartment assumptions, (5) poorly specified physiochemical activiation or deactivation metabolism parameters, (6) cellu lar and subcellular exposure pharmacodynamics, and (7) nonlinear parame ters. The simple models analyzed and solved above shall hopefully provide fundamental motivation and component building blocks for development of similar, but extended, analytic solutions, quasianalytic approximations or general numerical algorithms for more complex linear and nonlinear P B P K models that would be of practical use in risk analysis, especially Q R A and ED applications. Such issues will be explored elsewhere.


18. BROWN


317

References [1] Allen,B.C.; Crump, K.C.; Shipp, A . M . Risk Anal. 1988, 8, 531-544.


[2] Anderson, M.E.; Clewell III, H.J.; Gargas, M.L.; Smith, F.A.; Reitz, R.H. Toxicol. Appl. Pharmacol. 1987, 87, 185-205. [3] Blancato, J.E. Proceedings of the International Water Disenfection Con ference, Aug 31-Sept. 3, 1992; Int. Life Sci. Instit.: Washington, DC, in press. [4] United States Environmental Protection Agency. Federal Register, Part V, June 5, 1992 [5] Gear, C.W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice-Hall: Englewood Cliffs, NJ, 1971; pp.1-253. [6] Gibaldi, M; Perrier, D. Pharmacokinetics, 2nd Ed.; Marcel Dekker: New York, NY, 1982; pp 1-494. [7] Hirsch, M.W., Smale, S. Differential Equations, Dynamical Systems, and Linear Algebra; Academic Press: New York, NY, 1974; pp 1-358. [8] Kreyszig, E. Advanced Engineering Mathematics, 7th Ed.; John, Wiley: New York, NY, 1992; pp 1-1294. [9] Nakashima, E.; Benet, L.Z. J. Pharmacokin.Biopharm. 1989, 17, 673685. [10] Travis, C.C.; White, R.K. Risk Anal. 1988, 8, 119-125. [11] Travis, C.C. Risk Anal. 1990, 10, 317-321. RECEIVED

May 19, 1993


Biomarkers of Human Exposure to Pesticides - American Chemical

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