Below are the first 10 and last 10 pages of uncorrected machineread text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapterrepresentative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 25
PART 11
Mathematical Modeling
OCR for page 25
OCR for page 25
Modeling: An Introduction
Ellen ]. O'Flaherty
Fick's First Law states that the rate of diffusion of a solute down a
concentration gradient is proportional to the magnitude of the gradient:
dM dC
d = —DA dx'
(1)
where M is mass, C is concentration, D is the diffusion constant with
dimensions distance2/time, A is the crosssectional area of the diffusion
volume, and dx is the distance over which the infinitesimally small con
centration difference dC is measured. When Fick's First Law is restated
for diffusion across a membrane barrier of thickness dx, the concentration
gradient dC is approximated by the concentration difference across the
membrane, Car —C2, and DAIRY is the firstorder transfer constant kit for
diffusion across the membrane, with dimensions distance3/time, or vol
ume/time:
dt MIX ~ ~ 2)
= —kit (C~ —C2)
(2)
Fick's Law states that transfer of freely diffusible molecules across a
membrane should be first order. A large body of experimental observations
supports this interpretation of the kinetic nature of diffusion. There are,
of course, exceptions: Excretion from liver or kidney may not be first
order, and gastrointestinal absorption may take place by active processes
27
OCR for page 25
28 ELLEN J. O'F~HERh
C
FIGURE 1
Lit

for a few chemicals. But, in general, it is reasonable to assume as a
working hypothesis that absorption and distribution of exogenous chem
icals are first order. The rate of diffusion is dependent on the partition
coefficient and molecular size and configuration of the chemical, and on
its degree of ionization.
Thus, diffusion out of a single compartment is a firstorder process
whose rate constant, k', depends both on the chemical and on the tissue
and has the dimensions volume/time. If both sides of Equation 2 are
divided by V so that it expresses the rate of change of concentration, not
of mass since concentration is what is measured in viva then the rate
constant becomes the elimination rate constant ke' with dimensions time
(Figure 11.
dC
= —keC.
dt
(3)
Equation 3 is integrated to obtain the familiar firstorder expression for
C as a function of t,
C C  ket
(4)
Equation 4 has a single exponential term so that if the natural logarithm
of the concentration is plotted against time, the graph takes the form of
a straight line whose slope is —ke and whose ordinate intercept is the
logarithm of CO. The halflife is estimated from the value of ke' and the
volume of distribution from the dose and the value of CO. The model is,
of course, the onecompartment body model with firstorder elimination.
The body is not a single physiological compartment, however, and
rarely behaves as if it were a single kinetic compartment. More sophis
ticated models of the body are created by the addition of peripheral com
partments. The essence of different approaches to modeling lies in how
OCR for page 25
Modeling: An Introduction 29
these compartments are defined and in what kinds of variables mea
surements of concentrations or amounts, or values of physiological pa
rameters are used to drive the development of a quantitative model.
In the 1940s and 1950s, it was recognized that concentration behavior
in the central compartment of a multicompartment model could be rep
resented by a sum of exponential terms like the single term describing the
onecompartment model, one for each compartment in the model. Thus,
for any model with more than one compartment, there will be more than
one term, and the dependence of the logarithm of the concentration in the
central compartment on time cannot be linear. Instead, it takes a curvilinear
shape with a terminal straightline portion. By a process variously known
as feathering, peeling, or the method of residuals, curvilinear plots of in
C versus t were resolved into their component exponential terms. As many
terms were included in this feathering process as were required to account
fully for the curvature of the data. Such fits are now, of course, carried
out by nonlinear regression computer programs, but there was a time when
they were not.
In early modeling applications, model compartments were taken to have
exact physiological correlates. Because of the correspondence between
the number of compartments in a model and the number of terms required
in the equation describing the model, the number of exponential terms
necessary to account for the curvature in the data was taken to represent
the number of distinguishable exchanges between the central compartment
and peripheral tissues or organs, plus one term roughly equated with
wholebody loss.
An example of this approach appears in a paper on the kinetics of the
rapid phases of plasma free cholesterol turnover (Porte and Havel, 19611.
Free cholesterol labeled with i4C was incorporated into plasma lipoproteins
in vitro and administered to dogs by intravenous injection. Resolution of
the entire free cholesterol curve by successive subtraction of each com
ponent that is, by feathering gave five exponential terms with half
times of 4 min. 30 min. 65 min. 7 h, and 96 h. Porte and Havel compared
these halflives with turnover times reported for different pools of cho
lesterol, and concluded that the slowest, 96h component represented
metabolic turnover plus equilibration with very slowly exchanging com
partments; the 7h component represented formation of plasma ester cho
lesterol; the 65min component represented exchange of free cholesterol
between plasma and red blood cells; and the 30min component represented
exchange of free cholesterol between plasma and liver. The most rapid
component, with a halflife of 4 min. could not be related to any known
physiological compartment.
It quickly became apparent that forcing such a rigid correspondence
between exponential terms and physiological compartments generated a
number of problems, two of which are illustrated by the cholesterol ex
OCR for page 25
30 ELLEN J. O'FLAHERTY
cl
i ~
k21
k12
FIGURE 2
He
ample. It was not always possible to identify physiological correlates of
exponential halflives, particularly the shorter ones. And often more than
one process for example, metabolism and slow exchange presented
themselves as candidates for the source of an exponential term.
With general dissemination of explicit mathematical solutions of mul
ticompartment models and recognition of the implications of these solu
tions, in the mid 1960s a reaction set in. If there is only one compartment,
the halflife is the halflife of elimination, and dose/CO is the physiological
volume of distribution. When there is more than one kinetically distin
guishable compartment, the slopes and intercepts of the successive linear
segments that are revealed by the curvepeeling process are expressed in
appropriate units for calculation of halflives and volumes of distribution.
But these are not halflives that are descriptive of a single process, nor
are they physiological volumes of distribution.
The reason is apparent on consideration of the twocompartment model
shown in Figure 2 and below.
OCR for page 25
Modeling: An Introduction 31
dt = k2~C2—(kit + ketch, and
dC2 = kick —ketch.
(5)
Simultaneous integration of these two equations gives the explicit so
lution of the twocompartment model, in which the intercepts AD and Be
and the kinetic rate constants cx and ~ are expressed in terms of the rate
constants kit and kin for transfer between compartment 1 and compartment
2, the elimination rate constant ke, and the volumes of the compartments:
Car = AOeat + BOe~t
D(or—ken)
° Vat (a  Fj
B = D(k2~ — ~)
Vat (a  P)
= 112~(kI2 + k21 + ke) + t~kI2 + k21 + ke)2  4k2Ike] }/2}
,13 = 112~(k~2 + k2~ + key
[(kl2 + k21 + ke)2  4k21ke] 1/2}. (6)
Alpha and ~ are functions of all of the rate constants, and Ao and Be
are functions of the volumes as well as of the rate constants. Being hybrid
constants, they need have no direct physiological significance, although
of course they reflect the biochemical and physiological basis of the chem
ical's disposition. Consequently, the volume of distribution (calculated as
dose/B0) and halflife (calculated as in 2/~) need have no physiological
correlates.
Other apparent volumes of distribution can also be calculated. All are
constants that relate a concentration to an amount under a particular set
of conditions. But because kinetically determined volumes of distribution
usually do not correspond to real volumes of distribution, it became
commonplace in the 1970s to consider them simply as proportionality
constants.
The apparent volume of distribution is a useful pharmacokinetic parameter that
relates the plasma or serum concentration of a drug to the total amount of drug
in the body. Despite its name, this parameter usually has no direct physiologic
meaning and does not refer to a real volume (Gibaldi and Perrier, 19751.
This philosophy of modeling was, of course, in some respects a reaction
to what was correctly perceived as unproductive and in some cases mis
leading data interpretations as a result of insistence on too exact a cor
OCR for page 25
32 ELLEN ]. O'F~HERU
respondence between the terms of the equation describing loss of the
chemical from the blood and the physiological nature of compartments
organs and tissues exchanging with the blood. During this period also,
it became commonplace to minimize the significance of halflives derived
from any but the terminal slope of the plasma or blood concentration
curve. The terminal slope is used in the calculation of the biological half
life, which is generally accepted as an index of the persistence of the
chemical in the body.
The impetus for physiological modeling arose independently of classical
pharmacokinetics, and physiological modeling coexisted with classical
pharmacokinetic approaches during the 1960s and 1970s. In the 1980s,
it is beginning to emerge as the preeminent approach to pharmacokinetic
modeling. Because physiologically based pharmacokinetic modeling has
received so much recent attention, it is important to make the point that
physiological pharmacokinetics and classical pharmacokinetics are not
fundamentally incompatible. Although the philosophy behind the two ap
proaches is different and dictates their application for different purposes,
there is a direct link between the two approaches.
Let us return for a moment to Fick's First Law. Fick's First Law
describes the change in amount of a chemical with time. Thus, in the
closed twocompartment model, the rates of transfer across the membrane
separating the two compartments are expressed in terms of mass and kinetic
rate constants, or of concentrations and transfer constants or clearances:
= —ki2M~ + k2iM2
dt
= —kl2V1 C1 + k2l V2C2
(7)
At steady state, when dMlldt = 0, kl2VlCl = k2lV2C2; and since Cl =
C2, kl2Vl = k2lV2 = kt. The equality applies, of course, not only at
equilibrium but at disequilibrium as well. Thus,
(h —k,fC1 —C2)
(8)
It is not coincidental that the transfer constant of classical pharmaco
kinetics has the dimensions of a flow rate. In the referent fluid volume
for that flow rate lies the link between classical and physiological pharma
cokinetics. If transfers are perfusion limited that is, flow limited or first
order then the transfer constant is the rate of blood flow to the tissue
In the twocompartment closed model, let compartment 1 be the blood
subcompartment and compartment 2 the fluid subcompartment of a tissue.
Then, k~2V~ is the rate of blood flow to the tissue, which in this model
is equal to total blood flow since there is only one peripheral compartment.
OCR for page 25
Modeling: An Introduction 33
Substituting blood flow rate for knave and knave, we find that the rates of
transfer out of the blood and into the tissue fluid are expressed as blood
flow rate times the concentration difference:
dMi V dCI _n (r _
dt ~B\~1 — ~2J
(9)
With a single refinement, this is the fundamental equation of physio
logical pharmacokinetics. The refinement takes into account the fact that
chemicals do not simply equilibrate between body fluids but, depending
on their physicochemical properties, may be bound to tissue macromol
ecules or incorporated into tissue lipids. Thus, what is measured experi
mentally when a tissue is sampled is not C2 but Cal, the concentration in
the tissue including fluid subcompartment V2 and bound or sequestered
chemical. Because transfer is assumed to be flow limited, the concentration
of the chemical in efferent blood from the tissue should be equal to its
concentration in the fluid subcompartment of the tissue. Equilibration of
the chemical between the fluid subcompartment of the tissue and its bound
or sequestered forms is assumed to be very rapid, so that the partition
coefficient R = CIT/C2 = C/concentration in efferent blood) can be
measured and used to obtain C2 at any time from a measurement of tissue
concentration: C2 = CHAIR. Often, the partition coefficient is determined
in a separate in vitro vial equilibration experiment. Substituting CHAIR for
C2 in Equation 9, we obtain an expression for the rate of change in the
amount of the chemical in blood or tissue as a function of blood flow
rate, partition coefficient, and momentary blood and tissue concentrations:
dMi _ \! dC
dt
~ 1 dt
QB (C 1 — CII/R) .
The form of this equation suggests that it should be possible to substitute
physiological values of flow rates and volumes, and values of partition
coefficients, in order to obtain predictive, physiologically based phar
macokinetic models. In fact, this is the fundamental relationship on which
such models are based. The expressions for all peripheral, nonelimination
tissues will be of this form, with QB replaced by blood flow to the tissue
in question. The equation for the blood will be more complex but is directly
derivable from the same kinds of considerations of flow and partitioning.
It will include contributions from major tissue groups characterized by
different perfusion rates, and it may include input rate or elimination rate
terms. The equation for the liver may also include terms for metabolism
or for input by absorption from the gastrointestinal tract. Some of these
terms, particularly those describing metabolism, may not be first order.
OCR for page 25
34 ELLEN J. O'F~HERTY
The difference between classical and physiological pharmacokinetic
models, then, lies not in how they are constructed but in how they are
driven. In classical pharmacokinetics, no effort is made to assign physi
ological correlates to model parameters. A compartment is simply defined
as a volume (strictly speaking, as a fluid volume) that is kinetically ho
mogeneous. It is generally recognized that only in a very few instances
are more than three exponential terms required to describe satisfactorily,
within the precision of the data, the behavior of a declining concentration
curve; most often, two suffice. This understanding has given rise to a
group of models in which the body is represented by a central compartment
and one or two peripheral compartments which may be "shallow" or
"deep"; i.e., they may exchange relatively rapidly or relatively slowly
with blood plasma. Such important concepts as volume of distribution,
biological halflife, clearance, integrated total exposure following a single
dose, and achievement of steady state during chronic exposure arise nat
urally from these classical models. Their utility for characterization of the
behavior of a chemical, and for comparison of its behavior with that of
other chemicals, is firmly established.
Classical pharmacokinetic models support certain kinds of extrapola
tions in particular, extrapolation to different exposure conditions, with
reasonable assurance. Capacitylimited or other nonlinear kinetic behavior
can be incorporated into classical pharmacokinetic models. A specific
advantage of the models is that because the kinetic characteristics of the
compartments of which they are composed are not constrained, a best
possible fit to a data set can be arrived at by varying the values of the
parameters. Best estimates of parameter values can be compared across
experimental conditions, treatments, or chemicals to establish whether
apparent differences (effects) are statistically significant.
This strength of classical pharmacokinetic models is also their greatest
weakness. Lacking a physiological or biochemical basis, the models can
not take into account intraspecies changes such as growth, sexual matur
ation, or aging, and cannot reliably be used in interspecies conversion of
pharmacokinetic data. The need for interspecies conversion of laboratory
animal data, in particular, has led to the development of physiological
pharmacokinetic models, in which the unspecified compartments of the
classical pharmacokinetic models are replaced by actual organs and tissues
with their known blood flows. Because tissue volumes, blood flow rates,
and enzyme activities can be varied only within physiological limits in
these models, the models are not fit to experimental data in the classical
sense. Instead, gross discrepancy between the predictions of a physiolog
ical model and experimental observation requires reformulation of the
model in such a way as to account for the observed behavior.
OCR for page 25
Modeling: An Introduction 35
In a sense, then, we have come full circle, from early insistence on
correspondence between exponential terms and identifiable plasmatissue
interchanges to recognition that those interchanges do indeed give form
to the plasma concentration curve, although not in the sense in which they
were originally believed to do so. Physiological pharrnacokinetic models
have tremendous potential, particularly for speciestospecies conversion
of doseeffect data. But classical pharmacokinetic models still have their
place. Specifically, they are amenable to statistical treatment and, thus,
to hypothesis testing, whereas the purely physiological pharmacokinetic
models are not as readily treated statistically.
 Both physiological and classical pharmacokinetic models have valid
applications today. Both are capable of predicting the dose delivered to
a target organ, within somewhat different limits. The assumptions on
which the physiological pharmacokinetic models are based make them
uniquely suited to crossspecies applications. On the other hand, the de
pendence of classical pharmacokinetic models on experimental measure
ment of concentration or amount makes them especially well suited to
examination of questions about mechanisms of effects that involve changes
in pharrnacokinetic behavior.
REFERENCES
Gibaldi, M., and D. Perrier. 1975. Pharmacokinetics. P. 175. New York: Marcel Dekker.
Porte, D., Jr. and R. J. Havel. 1961. The use of cholesterol4C '4labeled lipoproteins as
a tracer for plasma cholesterol in the dog. J. Lipid Res. 2:357362.
OCR for page 25
52 KENNETH B. BISCHOFF
BLOOD
100
y
E
 o
he
Cal
He
o
c,, 10
On

2
Dog
Human
_ ~
.g
·:

,
0 1/2 1
2
TIME, hours
FIGURE 6 Comparison of data with model predictions for thiopental.
For kidney:
VK ddCtK
= QK(CP
CK ~ CK
—J — kK —
(TIC)
OCR for page 25
Physiologically Based Pharmacokinetic Modeling 53
, rPLASMA
QL QG
LIVFR
G l TRACT
QG
r~ Blilarysecretlon ~ ~ ~ ~ Gut Absorp1 lon
—6~—Feces
Gut Lumen
KIDNEY
1 . I
Urlne
MUSCLE
L ~
FIGURE 7 Model for methotrexate pharrnacokinetics.
For liver:
1~ 1~
QK
QM
VL d = (QL — QG) (CP — R ) + QG (R R ) rO, (l ld)
where r0 = [KL(CLIRL)]I[KL + (CLIRL)]
For bile ducts:
\ /
~ d—t = ri1 —ri, (i = 1, 2, 3)
For gut tissue:
(lie)
dCG C CG 1 kC C,
dt G( P RG) [Z 4 (KG + C + bCi) (llf)
\
For gut lumen:
dCGL 1 dCi
= Y
dt 4 t~i dt
(1 la)
4 dt = r3—kFVGLCI—4(K C+IC + bCI),and (llh)
OCR for page 25
54 KENNETH B. BISCHOFF
—4 d— = —kFVGL(Ci~ — Ci)
4 (KG + Ci + bCi), (i = 2, 3, 41. (lli)
Most of the terms and their origins should now be clear. The new ones
are as follow. In Equation tic, the term kKCK/RK represents the renal
excretion of methotrexate, which is close to the glomerular filtration rate;
in Equation 1 lde, ri is the bile flow of the drug, with the liver secretion
being saturable at high doses (the three compartments in series represent
the actual tubular or distributed nature of the real system); in Equation
1 ifi, the motion down the gut lumen is modeled by four compartments,
and the absorption term has both a saturable and a nonsaturable component
that is important only for very high doses, presumably passive diffusion.
The various parameters in Equation 1 lai were either estimated from
the values given earlier or were independently measured in the case of
the complex secretion steps. Then, the pharmacokinetic behavior was able
to be predicted in mice, rats, dogs, monkeys, and man over a dose range
of 3,000, all with the same model and with a consistent set of parameters.
Two examples are shown in Figures 8 and 9. Thus, the details of this
model must be a reasonably faithful representation of the actual physio
logical and pharmacological events, and should be of aid in interpreting
results of experiments. Also, valid predictions of local drug concentrations
for various dosage regimens are possible.
The flow diagram in Figure 7 has a rather complicated configuration
because of the importance of the enterohepatic cycle in the methotrexate
pharmacokinetics. There was little direct metabolism of the drug, however,
so metabolism is not a major route of elimination; the ultimate excretion
was by the renal or the fecal route. Another example considers the opposite
extreme, in which a straightforward anatomic flow diagram is the basis
but the metabolism is dominant. Dedrick et al. (1972) have considered
the drug cytosine arabinoside on the basis of the compartments seen in
Figure 10. The sizes of the boxes in Figure 10 signify the relative im
portance of the various regions. Again, the same types of balances were
used, but with metabolic terms in each based on known enzyme kinetics
and levels. Figure 11 shows one prediction of the concentrations of cy
tosine arabinoside and its metabolite, uracil arabinoside.
A final reduction in the complexity of the models is possible when the
excretion/metabolism processes are relatively slow compared with the
intercompartment blood flows. In this case, the entire body has an essen
OCR for page 25
Physiologically Based Pharmacokinetic Modeling 55
~ 10
of
o
F
z 1.0
z
o
~ 0.1
o
I
t~. ~
tN
1 ~ .
\ ~






~ GL
o
K
0~01 _
0 60 120 180 240
MINUTES
FIGURE 8 Comparisons of data with model predictions for methotrexate.
Mice, 3 mg/kg. Abbreviations: GL, small intestine; L, liver; K, kidney;
P. plasma; M, muscle.
tially identical time response, and a onecompartment wholebody model
is useful. In terms of Equations 9 and 10 this implies that:
r1 <
56 KENNETH B. BISCHOFF
£
10
o
~ 1.0
CD
Ad
o
C'
0.1
a:
O
~ 0.01
_
\

 M
1 1 1 1
0 90 180
MINUTES
270 360
FIGURE 9 Comparison of data with model predictions for methotrexate.
Man, 1 mg/kg.
in pharmacology, although they are usually used with empirical param
eters, and therefore, no specific examples will be given.
DISCUSSION
Various special situations can require the use of combinations of all of
the types of mass balance equations given above. The examples showed
the type of reasoning used in several instances, although the flowlimited
case was used in all of them. It appears that this quite often gives a good
estimate of the overall drug concentrations throughout the body, even
though the drug may be membrane limited in certain specific organs.
Thus, a combination of the analyses illustrated in the examples, plus the
use of Equation 6 for the specific region, might be a reasonable scheme
for both the overall drug distribution and the details of, for example, tumor
uptake. The work of Dedrick et al. (1975) is in some ways an illustration
of this.
OCR for page 25
Physiologically Based Pharmacokinetic Modeling 57
FIGURE 10 Model for cytosine arabinoside pharmacokinetics.
E3
Heartl
Am_
:3 G.l. Tract
,
_ L l
E
,
, ~
Kidneys ~ Urine
_
Lean
4.04 I/min
n2d
0.35
1.10
0.18
1.24
0 93
. _ _
_
Finally, it should also be mentioned that the precise definition of the
various anatomic regions is often somewhat flexible and can depend on
the exact situation. For example, the major barrier to the transport of many
drugs is not the capillary membrane but the cellular membrane; in this
case, tissue can be defined as intracellular space, and blood can be defined
as vascular plus interstitial space. The important point is that the physi
OCR for page 25
5~3 KENNETH B. BISCHOFF
10
at
o

E
z
Cal
z
o
CD
AS
g 0.1
a
o
_)
\^
_

~ ARA  C + ARA  U

~A—C
1 1 1 1 1 1





1
0 20 40 60 80 100 120
TIME (minutes)
140
FIGURE 11 Comparison of data with model predictions for cytosine arabinoside and adenine
arabinoside.
ological and pharmacological information should be used in formulating
the model, so that the several goals mentioned in the introduction to this
paper might be achieved. The reviews quoted above provide many ex
amples of this; all of these are based on flow diagrams similar to those
shown in Figure 7 or 10, with appropriate modifications for the specific
drug.
OCR for page 25
Physiologically Based Pharmacokinetic Modeling 59
FUTU RE RESEARCH N EEDS
I would like to end with a brief discussion of my view of some of the
important future research needs. The first is the use of nonlumped tissue
region models, with spatial variation of concentrations, for certain critical
tissues. This could be important in tubulartype regions or thick, quasi
homogeneous regions where diffusion must be accounted for other than
across a thin membrane. Of course, this also includes more detailed de
scr~ptions of intracellular fluid, which are probably necessary to quantitate
truly the biochemical drug effects. The above types of reactiondiffusion
models lead to partial differential equations, which are more difficult
mathematically, but with modern computing technology they should not
cause severe problems in calculations. A related issue is the use of more
realistic descriptions of the local blood flows in the microcirculation. This
is not a totally new area, of course, because the Krogh tissue cylinder has
been used for many years in physiology to model oxygen transport, and
two papers in this volume are concerned with a very detailed distributed
model of the lung to model transport and reaction of ozone (see J. H
Overton, lit. C. Graham, and F. J. Miller and F. J. Miller, J. H. Overton,
R. C. Graham, E. D. Smolko, and D. B. Menzel in this volume). A sort
of middle ground model combining lumped compartments with distributed
regions where needed was used by Flessner et al. (1984, 1985) to study
peritonealplasma transport, and by Morrison and Bedeck (1986) to study
the transport of cisplatin in the brain. The papers by Roberts and Rowland
(1986) mentioned above also report comprehensive studies and refer to
previous papers, one of which was an early approach by Pang and Rowland
(1977) for hepatic metabolism. All this work is quite recent and has not
yet been incorporated into very many pharrnacokinetic studies.
A second broad area is the addition of more realistic biochemical rate
expressions that involve known pathways and the like. This will virtually
always lead to nonlinear terms, and so simple mathematical solutions will
no longer be feasible. Finally, we must move forward with the modeling
of drug effects with the same type of fundamental philosophy and combine
these improved pharmacodynamic models with the pharrnacokinetics to
provide simulation of the actual problem: improved knowledge of tissue
levels at the site of action for improved risk assessment. A few examples
of this are discussed by Bischoff (19731.
REFER ENCES
Adolph, E. F. 1949. Quantitative relations in the physiological constitutions of mammals.
Science 109:579585.
Bellman, R., R. Kalaba, and J. A. Jacquez. 1960. Some mathematical aspects of che
motherapy. Bull. Math. Biophys. 22:181190.
OCR for page 25
60 KENNETH B. BISCHOFF
Bischoff, K. B. 1967. Applications of a mathematical model for drug distribution in mam
mals. In Chemical Engineering in Medicine and Biology, D. Hershey, ed. New York:
Plenum.
Bischoff, K. B. 1973. Pharmacokinetics and cancer chemotherapy. J. Pharmacokinet. Bio
pharm. 1:465480.
Bischoff, K. B. 1975. Some fundamental considerations in the application of pharmaco
kinetics to cancer. Cancer Chemother. Rep. Part 1 59:777793.
Bischoff, K. B., and R. G. Brown. 1966. Drug distribution in mammals. Chem. Eng.
Prog. Symp. Ser. 66:3345.
Bischoff, K. B., and R. L. Dedrick. 1968. Thiopental pharmacokinetics. J. Pharm. Sci.
57:13461351.
Bischoff, K. B., and R. L. Dedrick. 1970. Generalized solution to linear, twocompart
ment, open model for drug distribution. J. Theor. Biol. 29:6383.
Bischoff, K. B., R. L. Dedrick, D. S. Zaharko, and J. A. Longstreth. 1971. Methotrexate
pharmacokinetics. J. Pharm. Sci. 60:11281133.
Chen, H.S. G., and J. F. Gross. 1979. Physiologically based pharmacokinetic models for
anticancer drugs. Cancer Chemother. Pharm. 2:8594.
Dedrick, R. L. 1973a. Animal scaleup. J. Pharmacokinet. Biopharm. 1:435461.
Dedrick, R. L. 1973b. Physiological pharmacokinetics. J. Dyn. Syst. Meas. Cont. Trans.
ASME Sept.: 255257.
Dedrick, R. L., and K. B. Bischoff. 1968. Pharmacokinetics in applications of the artificial
kidney. Chem. Eng. Prog. Symp. Ser. No. 84, 64:3244.
Dedrick, R. L., and K. B. Bischoff. 1980. Species similarities on pharmacokinetics. Fed.
Proc. 39:5449.
Dedrick, R. L., D. D. Forrester, and D. H. W. Ho. 1972. In vitroin vivo correlation of
drug metabolism deamination of 1~Darabinofuranosylcytosin. Biochem. Pharmacol.
21: 116.
Dedrick, R. L., R. L. Zaharko, R. A. Bender, W. A. Bleyer, and R. J. Lutz. 1975.
Pharmacokinetic considerations on resistance to anticancer drugs. Cancer Chemother.
Rep. 59:795804.
Flessner, M. F., R. L. Dedrick, and J. S. Schultz. 1984. A distributed model of peritoneal
plasma transport: Theoretical considerations. Am. J. Physiol. 246:R597R607.
Flessner, M. F., R. L. Dedrick, and J. S. Schultz. 1985. A distributed model of peritoneal
plasma transport; analysis of experimental data in the rat. Am. J. Physiol. 248:F413
F424.
Gerlowski, L. E., and R. K. Jain. 1983. Physiologically based pharmacokinetic modeling:
Principles and applications. J. Pharm. Sci. 72:11031127.
Gibaldi, M., and D. Perrier. 1982. Pharmacokinetics, 2nd ed. New York: Marcel Dekker.
Gillette, J. R. 1985. Biological variation: The unsolvable problem in quantitation extrap
olations from laboratory animals and other surrogate systems to human populations.
Banbury Report 19: Risk Quantitation and Regulatory Policy. Cold Spring Harbor, N.Y.:
Cold Spring Harbor Laboratory.
Himmelblau, D. M., and K. B. Bischoff. 1968. Process Analysis and Simulation. New
York: John Wiley & Sons.
Himmelstein, K. J., and R. J. Lutz. 1979. A review of the applications of physiologically
based pharmacokinetic modeling. J. Pharmacokinet. Biopharm 7:127145.
Jusko, W. J., and M. Gretch. 1976. Plasma and tissue protein binding of drugs in phar
macokinetics. Drug Metab. Rev. 5:43140.
Krasovskii, G. N. 1976. Extrapolation of experimental data from animals to man. Environ.
Health Perspect. 13:51 58.
OCR for page 25
Physiologically Based Pharmacokinetic Modeling 61
KrugerThiemer, E. 1968. Pharmacokinetics and doseconcentration relationships. Pro
ceedings of the 3rd International Pharmacological Meeting, Sao Paulo, Brazil. Pp. 63
113 in PhysicoChemical Aspects of Drug Actions, Vol. 7. New York: Pergamon Press.
KrugerThiemer, E. 1969. Formal theory of drug dosage regimens. II. The exact plateau
effect. J. Theor. Biol. 23:169170.
Leonard, E. F., and S. B. Jorgensen. 1971. The analysis of convection and diffusion in
capillary beds. Annul Rev. Biophys. Bioeng. 3:293339.
Lutz, R. J., R. L. Dedrick, and D. S. Zaharko. 1980. Physiological pharmacokinetics: An
in vivo approach to membrane transport. Pharmacol. Ther. 11:559592.
Morrison, P. F., and R. L. Dedrick. 1986. Transport of cisplatin in rat brain following
microinfusion: An analysis. J. Pharm. Sci. 75:120128.
Pang, K. S., and M. Rowland. 1977. Hepatic clearance of drugs. I. Theoretical consid
erations of a "wellstirred" model and a "parallel tube" model. J. Pharmacokinet.
Biopharm. 5:625653.
Rescigno, A., and G. Segre. 1966. Drug and Tracer Kinetics. Waltham, Mass.: Blaisdell.
Riegelman, S., and M. Rowland. 1968a. Shortcomings in pharmacokinetic analysis by
conceiving the body to exhibit properties of a single compartment. J. Pharm. Sci. 57:117
123.
Riegelman, S., and M. Rowland. 1968b. Concept of a volume of distribution and possible
errors in evaluation of this parameter. J. Pharm. Sci. 57:117123.
Riggs, D. S. 1970. The Mathematical Approach to Physiological Problems. Cambridge,
Mass.: MIT Press.
Roberts, M. S., and M. Rowland. 1986. A dispersion model of hepatic elimination, 1,2,3.
J. Pharmacokinet. Biopharm. 14:227260, 261288, 289308.
Shen, D., and M. Gibaldi. 1974. Critical evaluation of use of effective protein fractions
in developing pharmacokinetic models for drug distribution. J. Pharm. Sci. 63:1698
1703.
Teorell, T. 1937. Kinetics distribution of substances administered to the body. Arch. Int.
Pharmacodyn. Ther. 57:205240.
Wagner, J. G. 1971. Biopharmaceutics and Relevant Pharmacokinetics. 1971. Hamilton,
Ill.: Drug Intelligence Publications.
Wilkinson, G. R. 1975. Pharmacokinetics of drug disposition: Hemodynamic considera
tions. Annul Rev. Pharm. 15:1127.
Williams, R. T. 1974. Interspecies variations in the metabolism of xenobiotics. Biochem.
Soc. Trans. 2:359377.
OCR for page 25