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OCR for page 429
PART V11
Summary: Prospectives and
Future Directions
OCR for page 430
OCR for page 431
Prospective Predictions and Validations in
Anticancer Therapy
Jerry M. Collins
I NTRODUCTION
At first glance, the processes of anticancer drug development and en-
vironmental risk assessment may not seem to have much conceptual over-
lap. However, there is a strong common thread based upon the need to
make decisions regarding allowable human exposure limits. In both cases,
heavy reliance is placed upon interspecies toxicological comparisons.
Risk assessment is based upon both mathematical models and experi-
mental data. For example, the data might be the incidence of tumor
formation in rodents following controlled laboratory exposures to a toxin.
The role of the model is to predict the incidence of carcinogenesis in
humans under a variety of occupational and/or environmental exposure
conditions. The weakest link in this process is model validation. Due to
ethical considerations, it is not usually possible to administer precise
amounts of toxic chemicals to humans. If a human population develops
an unusual form of cancer, epidemiologic detectives might be able to trace
the source to a particular chemical. The human exposure data are estimated
retrospectively in whatever fashion possible, but the uncertainty in these
calculations is a major hurdle in quantitative analyses. Once a specific
chemical becomes suspect, it would be possible to do a set of quantitative
experiments in animals.
Even if we accept these examples with their imprecise estimates of
human exposure, the total data base for model validation is very small.
Yet a variety of needs forces us to accept these models as the basis for
431
OCR for page 432
432 JERRY M. COLLINS
major prospective decisions that have an impact upon the health of citizens
and the economic well-being of corporations and communities.
The preclinical toxicology phase of drug development shares some of
the same facets as the safety testing of industrial pollutants or other po-
tential environmental contaminants. For example, animals are used to
determine a lethal dose, such as the lethal dose for 10% of the animals
tested (LD~o). That estimate is then used to determine safety in humans.
Perhaps the single largest difference between the development of anti-
cancer drugs and the assessment of risks from environmental contaminants
is that direct experimental evidence is obtained in humans that can be
(rapidly) compared with data from animals.
The treatment of a life-threatening disease requires a rather different
set of risk-to-benefit decisions than considerations of maximally allowed
pollutants. The drugs used for the treatment of cancer have narrower safety
margins than those used for most other diseases. In general, the ratio of
a therapeutic dose to a toxic dose approaches unity.
DRUG DEVELOPMENT
Drug development consists of a progression of steps (Table 1) that starts
with the discovery of a new compound and ends with a clinical deter-
mination of therapeutic utility. To begin human testing, a safe starting
dose is needed. Establishment of a safe starting dose is one of the chief
functions of preclinical toxicology studies. As reviewed by Grieshaber
and Marsoni (1986), the current preclinical toxicology protocol for anti-
cancer drugs provides the basis for a safe starting dose, tailored to potency
in rodents. The human starting dose is 1/10 of the mouse ODD, expressed
on a milligrams/square meter basis. Prior to human testing, this dose is
confirmed in a second species.
After the starting dose has been evaluated in patients, subsequent doses
are escalated. Although there is always therapeutic intent when an anti-
cancer drug is given to patients, the major scientific goal of initial clinical
trials is to determine the acute, reversible toxicity. The endpoint of these
phase I trials is called the maximum tolerated dose, or MTD. The MTD
TABLE 1 Stages of Drug Development
Name
Function
Preclinical Discovery Random or planned
Screening Bioactivity
Toxicology LD,o; organ sites
Clinical Phase I Safety
Phase II Activity
Phase III Efficacy
OCR for page 433
Predictions and Validations in Anticancer Therapy 433
is used to establish the dose for more detailed efficacy studies in phase
II testing. The procedure used for dose escalation must achieve a balance
between the desire to escalate slowly enough to be safe and the desire to
escalate fast enough to be efficient. The most commonly used procedure
is known as the modified Fibonacci scheme (Goldsmith et al., 19751. The
initial escalation is rapid (100%, or doubling of the dose); subsequent
escalations narrow down until the 30-35% range is reached (Figure 11.
In summary, there are two areas of risk assessment that are encountered
in these early clinical trials: (1) selection of a safe starting dose, and
(2) choosing the rate of dose escalation.
A_
In
O
£ 3
_.'
~ O
_ ~
_ ~ ~
a)
Q
~ _'
2 -
0.7 -
0.5 -
0 0.1
O O 0 07
0.05
_
0.03
10
- Doxorubicin
5 - 6-MP
Daunomyctn
Thalicarpine
Acitnomycin D
Vincristine
AZQ
- AMSA
Teroxlrone x5 ~
Teroxirone
Dlhydro-AC
Carbo-Plat
N—Methyl—For
Hon~oHar
ThioTEPA Triciribine
- AnguidTne
Triciribine x5 (I
~ — F-Ara-AMP— — Entry
- F-Ara—AMP x5
30-352;
30-35%
30 - 35~;
30 - 357
30-352;
30-35%
30 - 35%
30 - 357
30-35%
40%
50%
67~;
1 00%
FIGURE 1 Interspecies toxicity comparison. For these 17 anticancer drugs, the median MTD
in humans was equal to the mouse LD~o, when both doses were expressed on a milligram/square
meter basis. To compare toxicity on a milligram/kilogram basis, the ordinate was multiplied by
0.083. All drugs were given as single doses ( x 1) or as five daily doses ( x 5). Data in the
second column are from Grieshaber and Marsoni (1986). Data in the first column were collected
from information in the literature or on file in the Toxicology and Investigational Drug Branches,
Division of Cancer Treatment, National Cancer Institute. As a reference, the modified Fibonacci
escalation steps are shown in the third column. Adapted from Collins et al. (1986).
OCR for page 434
434 ~ ERRY M. COLLl NS
COMPARISON OF HUMAN AND MURINE TOXICITY
How well does this strategy work? In Figure 1, the ratio of (human
MTD)/(mouse Log) is presented for a series of anticancer drugs (Collins
et al., 19861. First, it is worth noting that this particular collection of
interspecies toxicology data, although not exhaustive, probably exceeds
any comparable compilation for environmental contaminants or other hu-
man toxins. Rather than focusing on specific drugs at this stage, it is
helpful to get some appreciation of the range of variation. It could be
argued that there is considerable variation, even though the average is
quite reasonable. It might also be reasonably argued, however, that this
level of agreement is adequate for comparative purposes. A number of
factors provide motivation to probe further. For example, patient safety
might be improved by a better understanding of the sources of this vari-
ation. Also, the efficiency of early clinical testing might be raised if the
toxicologic variation could be related to measurable determinants, such
as plasma levels.
What are possible explanations for this variation in toxicity between
mouse and man? Table 2 lists three possibilities: (1) differences in drug
metabolism, elimination, and binding; (2) exposure time differences; and
(3) target cell sensitivity differences.
Elimination rates determine the drug exposure, or C x T. the area under
the concentration versus time curve. The concept of C x T with regard to
drug toxicity originated during World War I (Prentiss, 19371. German
pharmacologists observed that mustard agents are equally toxic whether
a high concentration is inhaled for a short time or a low concentration is
inhaled for a long time. The essential feature is that the CxT is the
determinant of effect rather than the absolute concentration itself.
Most toxicologists have presented their dosing information in terms of
milligrams/kilogram. Based upon the relationships between body surface
area and body weight (Freireich et al., 1966), it is possible to interconvert
dosing data between milligrams/square meter and milligrams/kilogram.
For mice, the relationship is 1 m2 = 3 kg. For humans, it is 1 m2 = 37
kg. Empirically, either set of units can be used to present raw data. The
use of body surface area has a distinct advantage, however, when toxicity
TABLE 2 Potential Explanations for Variation in Toxicity Between
Mouse and Man
1. Species differences in drug metabolism, elimination, and binding
2. Schedule dependency due to exposure time differences
3. Species differences in target cell sensitivity
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Predictions and Validations in Anticancer Therapy 435
comparisons are made across species. The physiological determinants of
elimination rates (such as glomerular filtration rate or organ blood flow)
tend to be highly correlated with body surface area. Thus, if the dose is
expressed in milligrams/square meter and elimination rates (milliliter/min-
ute/square meter) are identical in mice and men, then the drug exposure
(C x T) will be the same in both species at the same dose. On the other
hand, if the dose is expressed in milligrams/kilogram, the dose in humans
that produces equal C x T will be 1/12th the mouse dose, because a
correction must be made for body surface area differences. The factor of
12 is simply the ratio of body surface area constants, 37/3.
Freireich et al. (1966), Skipper et al. (1971), and Schabel et al. (1983)
have reported that with many anticancer drugs, toxicity observations carry
across species on a milligram/square meter basis, as long as schedules are
similar. They also were aware, however, that there are exceptions and
that more complete exposure parameters such as C x T or plasma phar-
macokinetics allow more useful comparisons of toxic or therapeutic re-
sponses from experimental and clinical studies.
Doxorubicin appears to be an example of metabolism/elimination dif-
ferences. The MTD in man is fivefold greater than the LD~o in mice, on
a milligram/square meter basis. Yet, as shown in Figure 2, there is con-
siderable agreement between blood levels measured at equitoxic doses. It
appears that humans are more tolerant than mice due to a higher clearance
(milliliters/minute/square meter) for doxorubicin.
FACTORS OTHER THAN C x T
The second factor of possible importance is a difference in exposure
times. For some drugs, there are threshold concentrations or time depen-
dencies that are related to toxicity and/or mechanisms of action. For drugs
with equal clearance values in mice and humans (milliliters/minute/square
meter), Skipper and colleagues (1971) have made the point that a bolus
dose of equal milligrams/square meter generally produces rather different
time courses in mice and man (Figure 3~. If there is a threshold for action
and a critical exposure time, where the threshold lies can give major
differences in species response. For example, if the threshold in Figure
3 is set at 10-6 M, there is no effect in man. If the threshold is set at
10-7 M, however, the duration of effect is much longer in man than in
mice. Note that the time course for doxorubicin (Figure 2) is an exception
to the generalized pattern.
In a classic study by Quinn et al. (1958) 29 years ago, the threshold
effect was first demonstrated for the barbiturate hexobarbital. After a
standard dose of 50 to 100 mg/kg was given to mice, rats, and rabbits,
OCR for page 436
436 JERRY M. COLLINS
10-5
o
_'
o
·_
10-6
mu
o
c 10-7
r ~
At
10-8
- 7
- t O Mouse LD1 0 (1 8 mg/sq.m)
t
· Human MID (90 mg/sq.m)
.
_ _
·. .
..
. .
l
l
24
6 12
TIME (hours)
FIGURE 2 Doxorubicin plasma concentrations in mice and humans at equitoxic doses. Human
data were scaled from a 75-mg/m2 intravenous (i.v.) dose. Mouse data were scaled from a 75-
mg/m2 dose given i.v. to CDFl mice. Reprinted with permission from Collins et al. (1986).
there was substantial variation in the drug effectiveness. The times to
awakening were 12, 90, and 49 min in mice, rats, and rabbits, respectively.
When the plasma pharmacokinetics of hexobarbital were investigated, it
was found that the three species exhibited rather different elimination
rates, or plasma half-times. At the time of awakening, however, the plasma
drug concentration was similar in all three species. Thus, this is an example
of a species difference in drug effect that is determined by pharmacokinetic
changes in exposure patterns. Studies in dogs did not give as clear a
pattern as for the other three species.
The first two reasons for the species variation in toxicology have been
oriented toward plasma pha~acokinetics. The third factor essentially
covers all explanations that are not related to the delivery of the drug to
the site of action. There can be differences at the cellular level that de-
termine species sensitivity. For example, if the drug needs to be activated
OCR for page 437
Predictions and Validations in Anticancer Therapy 437
inside the cell, there may be a species difference in activation capabilities.
Fludarabine phosphate (F-Ara-AMP) has been found to have the largest
difference in dose (10- to 30-fold) between the mouse LD~o and the human
MID. This discrepancy is apparently an example of species differences
in target cell sensitivity. The phosphate group on F-Ara-AMP makes the
drug readily soluble, but it is rapidly cleaved to F-Ara-A in viva. Within
5 min following administration, only the F-Ara-A form can be detected
in plasma. As shown in the plasma profiles following single doses of
F-Ara-AMP (Figure 4), there is no obvious plasma pharmacokinetic ex-
planation for the species difference in toxicity. In contrast to the situation
for doxorubicin (in which equitoxic doses produced similar plasma drug
concentrations), it can be seen from these data that the plasma concen-
trations of the circulating species F-Ara-A are considerably different.
Studies with bone marrow cultures in vitro indicate that human bone
10 ~5 ~ ~
Mouse-Man sample plot
Skipper Concept
10 -6
S:
o
·_
O 10-7
o
\
\
\
\
\
Mouse \\
~~i-------- Human
·..
\
\
\
. .
10-8 0 3 6 12
.. - ,
·. .
24
TIME (hours)
FIGURE 3 Idealized plasma concentrations in mice and humans following equal bolus doses
(milligrams/square meter). Assume that volume of distribution (liters/kilogram) and clearance
(milliliters/minute/square meter) are similar in both species. Adapted from Skipper et al. (1971).
OCR for page 438
438 ~ ERRY M. COLLI NS
1000 ~
lo: 100
o
._
Cat
o
~ 10
of:
1
1
Ah\
- 452\
\
- titmouse LD10, 2600 mg/sq.m
~ ~`
`
`^
- ~_¢, Human MTD, 260 mg/sq.m
1 ~
, .
0 240 480 720
TIME (minutes)
FIGURE 4 F-Ara-A plasma concentrations in mice and humans at equitoxic doses. Mouse data
are from Noker et al. (1983). Human data are from Malspeis (Minutes of the Phase I Working
Group, Bethesda, Md., June 13-14, 1983). Reprinted with permission from Collins et al. (1986).
marrow cells are intrinsically more sensitive to this particular compound
than are mouse marrow cells (C. Poston et al., unpublished data). Because
bone marrow suppression is the principal acute toxicity in viva, it appears
that the species differences are due to target cell differences for this drug.
SUMMARY OF DATA
C x T information is listed for 12 anticancer drugs in Table 3. For the
first nine of these drugs, the C x T ratio is a useful predictor of the relative
toxicity in mice and humans. For S-azacytidine, doxorubicin (as already
discussed), and teroxirone, the C x T ratio is far better than the dose ratio.
For the next five drugs on this list, the C x T ratio was also a reasonable
predictor, although it was about the same as the dose ratio. For thio-
TEPA, the C x T ratio was also an improvement over the dose ratio.
OCR for page 439
Predictions and Validations in Anticancer Therapy 439
TABLE 3 The Mouse as a Quantitative Predictor of Human Toxicity:
Comparison of Dose Ratio (milligram/square meter basis) and C x T
Ratio for Human MID to Mouse LD1o
Dose C x T
Drug ratio ratio
1. 5-Azacytidine 6.5 (1.1)
2. Doxorubicin 5.0 0.8
3. Teroxirone 4.3 0.8
4. Diaziquone (AZQ) l.O (0-7)
5. Indicine-N-oxide 0.9 0.6
6. Amsacrine (AMSA) 0.8 1.3
7. Deoxycoformycin 0.7 1.1
8. Tiazofurin 0.7 0.9
9. Thio-TEPA 0.4 1.0
10. PALA 2.8 3.3
11. F-Ara-AMP 0.1 0.1
12. Dihydroazacytidine 1.2 0.3
There are also some drugs for which the C x T ratio was not an effective
predictor of toxicity. For PALA, both the dose ratio and the C x T ratio
were overly conservative; i.e., humans were threefold more tolerant than
mice. A more serious case arises when man is less tolerant than mice.
Two such cases were found in our survey. F-Ara-AMP was discussed
above. Dihydroazacytidine was also a case in which man was less tolerant
than mice, due to severe chest pain. As pointed out by Grieshaber and
Marsoni (1986), it is not easy to pick up some toxicities in a mouse
toxicology study.
Thus, the use of C x T seems to be a useful starting point for under-
standing differences in toxicity between mouse and man, but it is not
completely accurate. Further work is ongoing at the National Cancer
Institute that is exploring the use of C x T data to adjust escalation rates
in phase I trials (Collins et al., 19861.
CONCLUSIONS
The development of new anticancer drugs generates a unique quanti-
tative data base that can be used for interspecies comparisons of toxicity.
In addition to serving as a collection of empirical toxicity data, the data
base can be used for the testing of hypotheses regarding the fundamental
determinants of toxicity. For example, the role of pharmacokinetics can
be probed. Finally, the insights gained from analyses of past experience
can be put to practical use in the form of improvements to the drug
development process.
OCR for page 460
460 DAN ~ EL KREWSK' ET AL.
I.:. _
Gas
Exchange
1
Lung
Metabolism
. . . ~ . .. . .; ... ......... . .. . .. ._
.... . ..... .% .... ...... .... .. . ....... ....................
..... - ............. ;;; ........ .... ..
. ..,... .; ... ·,., x ~ ... ~~ .. N. -., A---,, %,
~ ~~ RICHLY PERFUSED ~
: ~ . . . .. ... ... r ~
. . ' ~ Y. -::: ~ ~ ~ .~.; -: ·.~::$:'~ ::: ~ · -: :: :: .: ·:: ::-: t:::'·:::: :- ~ :.:: :.:
:: :~ SLOWLY PERFUSED :: ~ ~ ~
I G. 1. TRACT
FIGURE 9 A physiologic phaImacokinetic model for methylene chloride (Andersen et al.,
1987).
of the GST path. Nonetheless, on the basis of other biochemical consid-
erations, Andersen et al. (1987) concluded that the GST surrogate is the
most appropriate predictor of tumor incidence.
Based on the GST path, the delivered dose in the high-dose groups is
approximately 100 times higher in the inhalation study than in the drinking
OCR for page 461
PK Data in Carcinogenic Risk Assessment 461
water study, as compared with the tenfold difference indicated by using
the administered doses. This is due to the fact that the doses in the
inhalation study were all well above the level necessary to saturate the
MFO path, leading to a high rate of activity in the GST path (figure 10),
whereas those in the drinking water study were not.
An upper confidence limit on the risk at low doses for female mice can
be calculated by using robust linear extrapolation based on the delivered
dose data from the inhalation study in Table 4 to be 0.56 (g/liter/day) - I.
At low doses, the rate of the GST path is approximately 0.036 mg/liter/
day/ppm, so that the low-dose slope on the administered dose scale is 2.0
x 1o-s Pam-. Working directly on the administered dose scale, the
low-dose slope value would be 2.4 x 1O-4 ppm-~, which is 12 times
higher.
To extrapolate to humans, we follow the results of Andersen et al.
(1987) and assume that the GST surrogate has the same potency across
species on a body weight scale. Calculation of the amount of GST surrogate
formed in humans requires determination of the human values of the model
parameters by allometric scaling or, preferably, by experimental mea-
surement. In the present example, Andersen et al. (1987) calculated many
of the physiological constants and the rates of the GST path allometrically,
but determined the metabolic constants involved in the saturable MFO
pathway expenmentally. The calculation also requires specification of the
dosing regimen. At low doses, inhalation of methylene chloride for 6 h
150
as
-
J _
~ 100
-
a)
-
ce
o
50
3
CO
CO
C,
o
/
/
/
-
/
100 200 300 400
Administered Dose (ppm)
FIGURE 10 GST activity in the liver in mice due to 6-in/day exposure to methylene chloride.
OCR for page 462
462 DAN ~ EL KREWSK! ET AL.
daily results in GST rate per part per million of about 0.012 mg/liter/day/
ppm, one-third of the rate for mice. Thus, the 95% upper confidence limit
on the low-dose slope for humans is 6.7 x 10-6 ppm-i, which is 36
times less than the corresponding value for mice calculated on the basis
of the administered dose.
It is worth noting that this ratio does not apply to all dosing routes. For
example, when drinking water exposure is modeled as continuous infusion
to the liver, human GST activity per unit dose is 0.16 (mg/liter/day)/(mg/
kg/day). This is about 3 times higher than that of mice, rather than 3 times
lower as calculated above for inhalation exposure.
C. Chen and J. N. Blancato (this volume) used a similar physiological
pharmacokinetic model to assess the risks of exposure to perchloroethylene
(PCE). Their model had no lung metabolism compartment, because one
path in the liver was assumed to account for all metabolism and to produce
the active metabolite. Its rate was used as the dose surrogate. Species
equivalence on surface area, body weight, liver volume, and air concen-
tration scales were all considered. In each case, the use of the metabolized
dose produced estimates of human risks 5-10 times lower than calculations
based on the administered dose.
SUMMARY AND CONCLUSIONS
The process of carcinogenic risk assessment based on the results of
toxicological experiments conducted in the laboratory involves certain
assumptions, such as that of low-dose linearity when extrapolating from
high to low doses. By using a simple mathematical pharmacokinetic model
for metabolic activation in which the probability of tumor induction is
proportional to the delivered dose, it was shown with a computer simu-
lation that saturation effects in metabolic activation resulting in a curvi-
linear dose response can have an impact on estimates of low-dose risk
obtained by linear extrapolation. In particular, saturation of detoxification
processes can result in an appreciable overestimation of risk, whereas
saturation of activation processes can lead to some underestimation of
risk. The most accurate estimates of risk are obtained with a linear dose
response.
These effects were further evaluated by using existing data on formal-
dehyde and vinyl chloride monomer, using covalent binding to DNA in
the nasal mucosa and the level of metabolism in the liver as possible
measures of delivered dose, respectively. With formaldehyde, low-dose
risks predicted on the basis of delivered dose were a factor of 4 lower
than those obtained by using the administered dose level, because of the
nonlinear relationship between the delivered and administered doses within
the experimental dose range. Because VCM metabolism is nearly pro-
portional to the level of exposure at low to moderate doses, estimates of
OCR for page 463
PK Data in Carcinogenic Risk Assessment 463
low-dose risk based on either the delivered or administered dose scale
were virtually identical.
The dose delivered to the target tissue may vary with time, depending
on both the exposure regimen and metabolic activation. By using a simple
mathematical compartmental model based on linear kinetics, it was dem-
onstrated that exposure regimens with different dosing intervals having
the same systemic availability lead to different peak concentrations in the
target tissue. In the absence of bioaccumulation, however, it was noted
that under the multistage model of carcinogenesis, the area under the
concentration-time curve is a better predictor of carcinogenic risk than are
peak concentrations.
Carcinogenic risk assessment can also require extrapolations between
different routes of exposure and from the animal model used to humans.
This can be done with PB-PK models. With methylene chloride and
perchloroethylene, it was noted that estimates of low-dose risks in humans
using predictions of dose delivered to the target tissue based on such
models can be substantially lower than traditional estimates based on the
use of the administered dose level.
In conclusion, pharmacokinetic models can be used to obtain more
accurate estimates of risk at low doses through the use of the dose delivered
to the target tissue as a surrogate for the administered dose in toxicological
studies, particularly when the response of interest is roughly proportional
to the delivered dose. Predictions of the internal tissue dose for other
routes of exposure can be obtained by using PB-PK models, provided that
measurements of the physiological and biochemical constants associated
with the different routes are available. Extrapolation between species can
also be facilitated by using physiological models to predict the delivered
dose in the species of interest. In those cases in which all of the relevant
model parameters cannot be measured directly in humans, however, these
must be obtained by scaling the corresponding values in animals. This
approach to interspecies extrapolation is of most use when the dose-
response curves for animals and humans are comparable when expressed
in tees of delivered dose.
ACKNOWLEDGM ENTS
We thank Dr. Richard Reitz for kindly providing the data shown in
Figure 10, and for helpful comments on the original draft of this article.
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PK Data in Carcinogenic Risk Assessment 467
APPENDIX: THE MULTISTAGE MODEL AND
TIME-DEPENDENT DOSING
Under the multistage model, the probability of a tumor occurring by
. . . ~
time t Is given by:
where
H(t) = f O f ok
P(t) = 1 —expE—H(t)l,
. .
(A-1)
fo2 II Ai~t~dul . . . duk (A-2)
i=1
denotes the cumulative hazard function and Lift) is the transition intensity
function for stage i = 1, . . . k (Crump and Howe, 1984~. To accom-
modate exposure to a particular xenobiotic, the intensity functions are
often assumed to be linearly related to the dose date at time t, with
mitt) = al + bidets,
(A-3)
where (ai, bi > 04. For constant exposure data = d, this reduces to the
usual form of the multistage model with:
tk k
H(t) = II (ai + bid)
k! id
(A-4)
To explore the effects of time-dependent dosing under the multistage
model, consider the simplest case in which only the rth stage is dose
dependent. Because bi = 0 for i ~ r, we have:
H(t) = aLt) + beta do,
where
(A-S)
ante = tweak= iai~lk!,
bate = (brlar~aft), and
do = ELd(Urk)] = To dLu~wfu; r, k— r + 1, t~du. (A-6)
Here, Mu; r, k—r + 1, t) represents the marginal density of Urk, the
rth order statistic in a sample of size k from the uniform distribution on
to, tl. This density is given by:
ur-"t—u~k-~ (A-7)
(O < u < t), with B(, ~ denoting the beta function (see Murdoch and
Krewski, 1987, for further details). It follows from Equation A-S that do
represents that fixed dose which, if given continuously from time O to
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Representative terms from entire chapter:
target tissue
468 DAN ~ EL KREWSK! ET AL.
time t, would produce the same cumulative hazard at time t as the variable
dosing regimen date. It follows from Equation A-6 that the use of a constant
time-weighted average dose
d, = Orb dfu~dult
(A-8)
will be valid when k = 1, because Mu; 1, 1, t) = 1/t for O < u < t.
When k > 1, however, the use of do in place of d, does not in general
lead to the same cumulative hazard at time t as that which accrues under
the variable dosing regimen data.
When only one stage is dose dependent, it follows from Equation A-S
that the excess risk is proportional to d i. Because this function is a weighted
average of the values of dfu) over O < u < t, the ratio R of the excess
risk under variable dosing as compared with that based on a constant
average dose is at most the maximum value of the density Mu; r, k—r
+ 1, tJ divided by its average value lit. This ratio can be written as:
R = ~ ~ leak ~ '(!
PART V111
Perspectives
