[CANCER RESEARCH 46, 3876-3885, August 1986]
Treatment Sequencing, Asymmetry, and Uncertainty: Protocol Strategies for
Combination Chemotherapy1
Roger S. Day
Department of Biostatistics, Harvard School of Public Health, and Dana-Farber Cancer Institute, Boston, Massachusetts 02115
ABSTRACT
This paper summarizes a large number of simulations which relax the
"symmetry" assumptions in the Goldie-Coldman treatment model, which
was symmetrical with respect to two drugs considered for use together.
The results are that, under relevant violations of the assumptions, nonalternating treatment schedules frequently outperform alternation and
combination substantially. The magnitudes of these effects are at least
as large as the improvements made by shortening the time periods
between treatment changes. Three protocol design strategies derived from
these results are described: (a) when there is no knowledge of parameters,
empirical trials in search of the best schedule can be contemplated, if
patients in such trials have fundamentally "similar" tumors, in a special
sense given the name "pattern homogeneity"; (b) when minimal knowl
edge of cell kill parameters is available, the "worst drug rule" could
perform remarkably well. This strategy is contrary to much of current
practice, but a clear rationale is proposed; and (<•)
when detailed knowl
edge of tumor parameters is available for each individual, detailed mod
eling to predict optimal schedule promises great improvements in treat
ment outcome.
Additional considerations addressed include factors determining the
merit of these strategies, suggestions for new laboratory research, and
implications for future clinical chemotherapy research.
INTRODUCTION
Drug resistance is usually implicated in the failure of chem
otherapy to cure cancer. Tumors responding to an agent will
often regrow and manifest resistance to the initially successful
agent. The most common mechanism appears to be that cells
within a single tumor differ from each other in their sensitivity
to chemotherapeutic agents and pass these differences on to
daughter cells. These differences have been demonstrated in a
variety of settings to originate during tumor growth from errors
in cancer cell genome replication (mutations in a broad sense)
(1-3). The errors are unlikely for any single cell division but
nearly inevitable by the time the tumor is of diagnosable size.
[This type of variability, referred to as "tumor heterogeneity,"
affects other important characteristics as well: metastatic abil
ity, antigen expression, and morphology, for example (3-6)].
Goldie and Coldman (7, 8) have argued that this picture of
drug resistance supports the intuitively appealing strategy of
administering two active drugs concurrently when possible, and
in alternation when toxicity precludes the concurrent schedule,
in place of the more usual sequential schedule, switching from
one drug to the other just once. [Here the term "drug" is a
shorthand for any treatment modality to which heritable resist
ance, in the broadest sense, can develop. Thus the implications
were intended to extend to combined-modality therapies (9). In
addition, although their work focused on strategies for the use
of two drugs, the extension to more than two seems straight
forward.]
These authors' arguments rest on a mathematical model for
Received 9/23/85; revised 3/12/86; accepted 5/1/86.
The costs of publication of this article were defrayed in part by the payment
of page charges. This article must therefore be hereby marked advertisement in
accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
'Supported by USPHS Research Fellowship T32-ES07142-03, by National
Cancer Institute Research Grant CA-39640, and by a Biomedicai Research
Support Grant administered through Dana-Farber Cancer Institute.
tumor growth and treatment incorporating mutation to drug
resistance. Unavoidably, as with any mathematical model for a
complex biological process, this model is built on simplifying
assumptions which are known to be violated in reality. The
adequacy of assumptions can be investigated by relaxing them;
if plausible departures from the assumptions lead to major
changes in the conclusions from the model, great caution is
needed. In particular, if reality differs sufficiently from the
Goldie-Coldman assumptions, the concurrent-or-alternating
strategy may be bested by other strategies. (As an extreme
example, the benefit from combining or alternating with an
ineffective drug is nil at best.) Some important questions are
whether the Goldie-Coldman strategy is best in a wide variety
of clinical situations, what rules govern the exceptions, and
how information available on the cell kinetics and drug sensi
tivity of subpopulations should be used to choose treatment
schedules.
This paper addresses these questions. To extend the work of
Goldie and Coldman, parameters determining the process of
tumor growth and the effects of treatments were systematically
varied, relaxing the "symmetry" assumptions of the original
Goldie-Coldman model. Symmetry means, in part, that the two
drugs were equivalent in the sense that swapping one for the
other throughout the treatment schedule would lead to identical
results. (A more complete description of symmetry appears in
"Materials and Methods.") For each such choice of parameter
values, the cure probabilities achieved by each of 16 represent
ative treatment schedules were computed.
We relaxed assumptions in stages of progressively greater
departures from symmetry. At each stage we determined which
types of tumor kinetics and resistance patterns favor the alter
nating strategies, which ones favor interweaving but not strictly
alternating strategies, which ones favor one-drug strategies, and
which ones are indifferent. The performance of the optimal
strategy was compared with the performance of alternating and
one-drug strategies and with the schedule choice indicated by a
previous stage.
The results are that under relevant violations of the symmetry
assumptions, nonalternating treatment schedules frequently
outperform alternation and combination substantially. The rel
ative performance of schedules follows understandable patterns.
A simple strategy is introduced (the "worst drug rule") which
leads to profound improvements in outcome over a wide range
of parameter values. This strategy appears initially counterin
tuitive and in fact is contrary to current practice, but this paper
proposes an easily grasped viewpoint in which the strategy is
consonant with common sense.
Three potential protocol design strategies derived from these
results are described. When there is no knowledge of parame
ters, empirical trials in search of the best schedule can be
contemplated, as long as patients with the same tumor type
have similar parameters. Using minimal knowledge of cell kill
parameters, application of the worst drug rule could have
excellent results. Finally, an even larger improvement could
result from predicting the best schedule based on detailed
knowledge of the cell kinetics and drug kill parameters.
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TREATMENT
SEQUENCING,
ASYMMETRY,
The usefulness of these strategies is limited by two major
sources of uncertainty about the target tumor, each with its own
ramifications. The first is the difficulty of measuring the rele
vant parameters; as assay techniques improve, the last two
strategies above will become more attractive. The second type
of uncertainty arises from the possibility of fundamental differ
ences between the clinical tumors of patients who ostensibly
have the same disease; if these differences turn out to be small,
in a limited sense which will be called "pattern homogeneity,"
then the first two strategies become attractive. The role of
pattern homogeneity appears to deserve careful consideration.
These sources of uncertainty are explored in the "Discus
sion." Some implications for the future of clinical chemother
apy research and some suggestions for relevant laboratory re
search will be outlined.
MATERIALS AND METHODS
Outline of Approach. We conformed to the basic setup and nomen
clature of Goldie and Goldman (8) with respect to drugs and cell types.
There are two drugs, called A and B, and four cell types: sensitive to
both drugs (S); resistant only to drug A (RA); resistant only to drug B
(RB); and resistant to both (RAB). The tumor is assumed to begin with
a single sensitive (S) cell.
The doubling time of S-cells is taken to equal unity, which may be
thought of as 1 month. Treatment courses ("doses") are constrained to
be 1 month apart and are modeled as if the entire course and all
resultant cell death occurs at a single instant. Treatments are initiated
at 30 doublings of the S-cell subpopulation, when the tumor bulk is at
roughly 10' cells. Justification for these choices can be found in stand
ard references (10, 11).
The goal was to summarize the effectiveness of a large number of 2drug treatment schedules on a large number of 4-cell type tumor models.
(Herein "model" will refer to a particular assignment of values to the
parameters.) The problem was simplified by circumscribing the models
and schedules with a small number of carefully chosen representatives.
A central symmetrical model was defined by choosing medium values
for each parameter. High and low values were chosen for varying these
parameters in order to violate symmetry. A battery of 16 treatment
schedules was chosen to represent two important dimensions in the
problem of combining two treatments: the total number of courses of
each drug (the dose mix), and the timing for the first course for each
drug (the time mix). For each model and each treatment schedule, the
probability that no tumor cells remain at the end of treatment (the cure
rate) was computed. From this, a "comparative cure index" (cure index,
AND
UNCERTAINTY
resulting probability that no cells are present at the end of treatment
was computed. All results were computed conditionally on the event
that the number of tumor cells at the time of treatment start is nonzero;
this is necessary because a positive death rate was allowed, creating the
possibility that the tumor dwindles away before it is established. The
computations were performed using modification of the TREATS
program developed at Dana-Farber Cancer Institute and Harvard
School of Public Health.
Choice of Parameters. The parameters of the process are as follows.
The mutation rate, MR(X,\), is defined as the proportion of cell
divisions of X-type cells which yield one X-type and one Y-type cell.
The doubling time, DT(X), is defined as the time at which the mean
number of progeny of a single X-cell is 2. The death rate, DR(\), refers
to 1 minus the ratio of mean cell cycle time to doubling time for Xtype cells. Finally the log kill, tf(D,X), is defined as -log(probability
that an individual X-type cell is killed by a course of drug D).
Table 1, middle column, gives parameter values for a symmetrical
model. This model is symmetrical in the sense that the cell kill was the
same for each drug on its respective sensitive and resistant cells: /i(A,S)
= #(B,S); tf(A,RB) = AT(B,RA);tf(A,RA) = A"(B,RB);and AT(A,RAB)=
A(B,RAB). There is also symmetry in the kinetic assumptions;
growth rates and death rates for all cell types are identical, and corre
sponding mutation rates are identical: A/A(S,RA) = A/A(S,RB);
and A/A(RA,RAB) = A/A(RB,RAB). Back-mutation rates such as
MR(RA,S) are assumed to be zero. The models which we considered
are neighbors of this central model.
To move the model away from the symmetry assumption, we altered
the 16 parameters which enter asymmetrically in the model. These are
the top 16 listed in Table 1. When varied, these parameters were
allowed to take either the upper or lower values appearing in Table 1.
When fixed, they took the middle value.
The range for cell kill parameters was chosen by trial to encompass
most models for which the best cure rates would be between 0.01 and
0.90 (neither uselessly low nor ridiculously high) and resistant cells
would be the primary cause of treatment failure. MRs, DTs, and DRs
were chosen to span the range of measured values generously.
Whether the log kill range should represent mild or extreme differ-
Table 1 Model parameter
Notation: S, RA,
RA is resistant to A
The last four entries
All the nonspecified
for short) was computed for each schedule, comparing its cure rate to
the cure rate of the other schedules on the model under consideration.
Mathematical Model of the Growth-Treatment Process. A general
continuous-time stochastic birth-death multitype branching process
(12,13) was used to represent the time evolution of the number of cells
of each cell type in a heterogeneous cell colony. In this process, each
cell exists until, at a randomly chosen time, a cell kinetic event occurs,
resulting in the replacement of the given cell by some other set of cells.
The kinds of cell kinetic events are mitosis (replacement by two cells
of the same cell type), mutation (replacement by one cell of the same
type and one of a different type), death (replacement by no cells), and
conversion (replacement by a single cell of a different cell type). The
lifetimes of cells are assumed to be independently exponentially distrib
uted. Thus any pair of cells at a point in time have statistically
independent futures. A treatment course is assumed to result in the
death of each cell with a fixed probability determined by the cell type
and the drug used; the fate of each cell is independent of that of any
other. A detailed description of this process, as well as a general
computational solution, appears elsewhere (13). For this application,
our computational method varies from that of Goldie and Goldman
only in introducing a minor improvement in the way cell death is
handled.
For many treatment schedules and sets of parameter values, the
values tested
and RB, RAB are cell types; S is sensitive to drugs A and B,
only, RB is resistant to B only, and RAB is resistant to both.
are parameters entering symmetrically, and were not varied.
mutation rates were assumed to be zero.
ParameterDeath
ratesOA(RA)DA(RB)Doubling
timesO7HRA)D7-(RB)Mutation
ratesjW?(RA,RAB)MR(RB,RAB)A//?(S,RA)AH?(S,RB)Cell
killsAXA,S)AT(A,RB)AT(A,RA)tf(A,RAB)AT(B,S)tf(B,RA)tf(B,RB)A(H.K
VitiDoubling
timesDT(S)07*(RAB)Death
ratesDR(S)DÄ(RAB)Low0.00.00.250.25io-«io-'10-'io-«110.00.0110.00.0ValuesMedium0.900.901.01.0io
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TREATMENT
SEQUENCING,
ASYMMETRY,
enees is unclear. Is one concerned only about drug pairs that would
ordinarily be considered useful, or those which might be useful? Even
if the former is the case, however, do currently used drug pairs have
such large asymmetries? The state of the art in log kill measurement is
probably insufficient to settle the matter, since the critical unknowns
(as we shall see) are the kills of doses achieved on the in vivo resistant
subpopulations. It is not even clear whether, say, a range from 1 to 3
in log kill is mild or extreme. A range of 2 orders of magnitude in cell
kill may not be as large as it seems. In fact, patient variability alone
might generate uncertainty that large concerning a single log kill. Under
a linear dose-response assumption, this range would represent only a
3-fold difference between tumors in the drug concentration actually
achieved physiologically; such a degree of patient variability in drug
clearance and tumor perfusion is observed (14). These considerations
led to the choice of broad ranges for cell kill parameters, based on the
cause-of-failure criterion.
The kinetics parameter ranges perform a different role, as a potential
"spoiler." Rules for choosing treatment based on log kill values might
perform well under simplifying symmetry assumptions about the cell
kinetics but might be invalid in general. To detect this, it is appropriate
that the kinetics parameter ranges be fairly extreme.
"Mild" departures from the central model (models which differ from
the central model only in one parameter) were examined exhaustively.
Particularly interesting classes of more severe departures were also
studied, including the most extreme cases, in which all of the asym
metrical parameters differ from the central model.
Test Battery of Treatment Schedules. Attention was restricted to the
212= 4096 possible sequences consisting of 12 courses a month apart,
starting at month 30. Thus we fix time intervals between courses and
the total number of courses, varying only the choice of drug each
month. The number of courses, 12, was chosen to be long enough so
that treatment failure is generally due to drug resistance, and not to
premature termination of treatment.
To minimize computing and make the summary of results easier, a
representative battery of 16 treatment schedules was chosen. Table 2
shows the battery of schedules. These schedules vary in the number of
courses of each drug (0, 3, 6, 9, or 12), which we call the "dose mix,"
and the month of first course for each drug (1, 2, 4, 7, 10, never), which
we call the "time mix;" they are presumed to cover the interesting
issues of schedule choice.
Scale for Comparing Performance. A measure was needed for the
superiority of one schedule over another, that is, a comparison between
two "cure rates" (Ps). The logit scale
log,,
- P
(i.e., log odds of cure) was found to be appropriate.
The practical significance of a large logit difference between the cure
rates of two schedules depends on the actual (untransformed) / V As
an example, an improvement of 0.5 on the logit scale can represent an
improvement in cure rate from 0.10 to 0.26, or from 0.01 to 0.03, or
from 0.001 to 0.003. Thus an improvement of 0.5 or greater would be
clinically important for a tumor type which is occasionally cured, while
an improvement larger than 1.0 might be important even for a tumor
type which is currently virtually incurable.
The logit scale is desirable because it does not penalize a comparison
of two Ps if they are both small. In theory, the untransformed Ps
themselves would provid. the best scale for treatment policy purposes,
since the difference in Ps is proportional to "lives saved." In reality, the
increase in the fraction of cures achieved when an improved schedule
is used is dependent on the base line cure rate. This base line is sensitive
to some fairly arbitrary choices in the model setup (such as time between
courses and time of treatment initiation). Using the logit scale ensures
that our results are fairly robust to the model setup, although it requires
that the interpretation include considerations of base line.
To assign a single number to each schedule, representing its perform
ance relative to other schedules, a comparative cure index is computed,
equal to the average difference, on the logit scale, between P for the
schedule of interest and the Ps for the other two-drug regimens in the
test battery.
RESULTS
Results for the Central Model. For the central (symmetrical)
model the superiority of two-drug regimens over the one-drug
regimens was clearly demonstrated. (The probabilities of cure,
and their corresponding comparative cure index, are seen in
Table 3.) On the logit scale, the alternating schedules outper
formed one-drug regimens by 3.6, indicating that alternation
would raise the odds of cure by more than 3 orders of magnitude
in this example. They also beat the least symmetrical two-drug
regimens (9A3B, 9B3A) substantially, scoring higher by 0.80.
The average advantage of the alternating over the other twodrug schedules, the comparative cure index, was 0.33.
Among the schedules with equal total doses of each drug, as
one passes from sequential (6A6B) to alternating jABj, the cure
index improves from -0.04 to 0.33. (The strategy of interweav
ing more finely would have its ultimate culmination in the
schedule with simultaneous administration of A and B, at 2month intervals. This "mixture" schedule adds another 0.16 =
0.49 - 0.33 to the log odds of cure. To avoid assumptions about
independent action of A and B, mixture schedules are not
discussed further here.)
Thus, this work confirms the results of Goldie and Goldman
but also reveals that coarse interweaving of the two drugs
provides most of the advantage of the optimal schedule and
that finer interweaving provides relatively little additional im
provement.
Stage 1: Single-Parameter Variations. The central model was
altered by allowing each parameter, one at a time, to take on
Table 3 Comparative cure index for each schedule on the central model
Total
A later. B earlier -
A B
12
09
36
63
90
3B9A6A6B
I3ABI
|B3A|
I3A3BI
6B6A3A9B
|AB| |BA) |3B3A|
IA3BI
|3BA|
Index"
12A
Probability
(cure)
2.6 x IO-1
9A3B
I3ABI
|B3A|
3B9A
0.15
0.23
0.40
0.41
-0.47
-0.24
0.12
0.12
6A6B
I3A3B)
|AB|
Combination6
IBA|
I3B3AI
6B6A
0.32
0.44
0.52
0.62
0.52
0.44
0.32
-0.04
0.18
0.33
0.49
0.33
0.18
-0.04
3A9B
|A3B|
I3BAI
9B3A
0.41
0.40
0.23
0.15
0.12
0.12
-0.24
-0.47
Schedule
Table 2 Treatment schedules tested
Notation: A number prefixed before a drug name (A or B) indicates the number
of times that drug is repeated. Patterns in braces are repeated until 12 total doses
are given. Example: |3AB| represents the schedule AAABAAABAAAB.
_ A earlier, B later
AND UNCERTAINTY
-3.29
2.6 X IO"4
-3.29
12B
" Column 3, "index", is the comparative cure index: the average comparison
with all two-drug schedules, on the logic odds-of-cure scale.
* The combination schedule gives a dose each of both A and B at every other
9B3A12B
1212A9A3B
time point.
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TREATMENT
SEQUENCING,
ASYMMETRY,
its high and low values as given in Table 1. The results are
displayed in Fig. 1 and summarized in Table 4.
A word about the graphical technique in Fig. 1 is in order. It
has proven indispensable for displaying a large amount of data
in a way that makes it easy to notice patterns and exceptions.
It also provides a visual reminder of the dangers of drawing a
conclusion from a calculation based on a single set of acceptable
parameter values. Each line represents a comparative cure
index; horizontal lines are positive values, vertical lines are
negative values, and the length of line is proportional to the
absolute value of the index. Each group of 14 lines displays the
outcomes for the 14 two-drug treatment schedules, as calculated
for one model. The schedules are physically placed within the
group in analogy with Table 2.
In most cases, the two alternating schedules |ABj and {BAj
are optimal (these two differing little). There are two excep
tions.
The biggest failure of the alternating schedules occurs when
DT(RA) is low, so that the RA cells grow exceptionally quickly.
In this case, early doses of B are critical to treatment success,
although the total number of B doses is not so important. The
advantage of the optimum schedule, 9B3A, over the alternating
schedule (ABj, is a factor of 10 in increased odds of cure (a
difference of 1.0 on the log odds scale). The B doses must be
given early to control the growth of RA cells, which otherwise
would be the cause of treatment failure.
Another substantial change in performance occurs when
A(A.K B) is low. In this case, paradoxically, more doses of A
are called for. The best schedule is 3B9A, with (B3AJ a close
AND UNCERTAINTY
runner-up. The advantage over alternating schedules averages
0.66 in log,o odds. All schedules with 9 courses of A in total
outperform the schedules with 6 each of A and B. Among the
schedules with 6 each of A and B, early A is better (in keeping
with the paradox), but among the schedules with 9 of A and 3
of B, early B is better. Thus the total dosage is more important
here than the timing.
The term "worst drug rule" will refer to any strategy choosing
a schedule with more or earlier doses of the drug with lower
cell kill. The term was chosen with tongue in cheek, since it has
some exceptions as we shall see. Nevertheless, it has important
implications. The worst drug rule will be seen to be counter to
much of current clinical practice. With a bit of thought, the
reason it works becomes obvious. Two jobs must be done to
effect a cure: the elimination of RB cells by drug A; and the
elimination of RA cells by drug B. If A is weaker, more (or
earlier) doses of A will be needed in order to control the
development of RB cells.
The pattern of optima laid out in Table 4, Column 3, might
be regarded as a refinement of the alternating rule, to be used
when qualitative information about a single asymmetry is avail
able. Accordingly the strategy of choosing a schedule according
to this pattern will be called the "stage 1 rule," and its validity
will be examined under wider departures from symmetry.
Stage 2: Variations in Log Kill Values. In stage 2, the eight
log kills were varied in all combinations, subject to equalities
representing lack of cross-resistance:
= A"(A,RAB),
Central Model
Scale
Comparative Cure Index
= deviation from average
in log (odds of cure)
— = 1.0
,
MR(S,RA)
= -1.0
MR(RA,RAB)
DT(RA)
DR(RA)
Fig. 1. Pictographs representing cure indices for
each two-drug schedule: central model and singleparameter variations. Each pictograph, a cluster of
whiskers in 3 rows, displays the cure indices for 14
treatment schedules. Within each pictograph, the
schedules are laid out as in Table 2. Schedules in the
top row have more As in the dose mix, the bottom
row has more Bs, the left side has B introduced late,
and the right side has A introduced late (rows slanted
up for clarity). Horizontal whisker, positive cure
index for the corresponding schedule, with value
proportional to length of whisker; vertical whisker,
negative cure index. Examples of schedules which
outperform alternation are circled. Parameter values
for these models are given in Table 1.
Varying
Kinetic
Parameter*
low value
high value
K(A,S)
K(A,RA)
K(A,RB)
K(A,RAB)
Varying
Cell-Kill
Parameters
low value
'.
1 '
high value
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TREATMENT
SEQUENCING,
ASYMMETRY,
Table 4 Results from stage I (parameters varied singly)
See Table I for parameter abbreviations.
Notation: +, parameter takes the high value in Table 1; —,parameter takes
the low value in Table 1. The comparative cure index, the average comparison
with all two-drug schedules (on the logio odds-of-cure scale), is tabulated for the
optimum schedule. Arrows indicate poor performance by alternating schedules.
Comparative cure index
AND UNCERTAINTY
relative to other 2-drug schedules, is greater than for the central
model when AT(A,R)and tf(B,R) take their low value (zero). If
the kill on sensitive cells is high, this is associated with a strong
requirement for a good "time mix"; i.e., both drugs must be
given early. If the kill on sensitive cells is low, however, it is
the "dose mix" which must be good; i.e., there must be six
OptimumI3A3B)+
ParameterMR(S,RA.)A/Ä(RA,RAB)O71(RA)DR(RA)K(A,S)AT(A,RA)JÕ(A,RB)Af(A.RAB)Change
doses of each
drug, but timing is not critical.
Examples of the worst drug rule are seen for the asymmetrical
I3B3AII3A3BJ+
models. The workings for one example are displayed in detail
in Fig. 3. The schedules whose consequences appear there are
IBAI9B3A+
0.690.310.610.270.410.330.780.21->
those circled in Fig. 2, with the model parameters as indicated
I3A3BI¡AB)+
in Fig. 2, namely: high AT(A,S)and low #(B,S), AT(A,R),and
K(B,R). K(A,S) is larger than #(B,S), yet the schedules with
IBA)|AB|/|BA|+
more doses of B, 3A9B and |A3Bj, perform better. The alter
IABI/IBAIIBAI+
nating schedule performs poorly: Pr(cure) = 0.007. Failure is
IABI3B9A+
due to overgrowth by RA cells. The schedule 3A9B produces
0.980.270.290.58IABI0.310.170.460.08-0.320.310.610.160.410.330.730.170.420.270.260.58|BA|0.170.310.320.24-0.090.170.500.270.41
9% cures. RA cells are eliminated; failue is due to RAB cells
IABI/IBAIIBAI+
present at treatment start and RAB cells from RA cells in the
IABIOptimum0.310.310.460.24-» first 3 months of treatment. [In this instance, this schedule
corresponds roughly to both the stage 1 rule and a strategy of
"use the best drug until failure, then switch." However, this
Using the equality assumptions to simplify notation, K(A,S)
represents A"(A,RB) as well as itself, and we use A(A,R) for correspondence is fortuitous; for example, if one raises /f(A,R)
and tf(B,R) from zero to 0.25, the regrowth of RA cells is
both /f(A,RA) and /f(A,RAB), and of course likewise for drug
slowed enough to extend the "complete response" to 12 months,
B.
Results appear in Fig. 2, with a summary in Table 5. In all but the cure rates and therefore the comparative performance
cases, the alternating schedules either win over the one-drug
of schedules change little.] The best schedule is (A3B), with a
20% cure rate. It has sufficient B to control the RA cells and
schedules substantially or else lose by small margins.
Of the 16 models, 4 are still symmetrical. All of these favor introduces the B early enough to limit further development of
the alternating regimens, but by differing amounts and for RAB cells from RA cells.
different reasons. For example, the advantage of alternating,
A more dramatic case of a nonalternating schedule doing
Scale
= 1.0
K(A,R)
= -1.0
low
high
low
K(B,R)
low
Fig. 2. Pictographs representing cure indices for
each two-drug schedule: stage 2, variations in cell
kills. See Fig. I for an explanation of the represen
tation. Cell kill values are from the "Low" and
"High" columns of Table 1 and are subject to equal
ities representing absence of cross-resistance. Sched
ules 12A and 12B are included if their cure index is
positive. Examples of the worst drug rule are found
here. For example, in the four models in the upper
right, where AT(B,S)< tf(A,S), schedules with more
B doses are generally favored (with one exception).
The three circled whiskers represent simulations set
out in more detail in Fig. 3.
^=-
high
K(B,S)
— _. t
high
low
high
K(A,S)
3880
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TREATMENT SEQUENCING, ASYMMETRY, AND UNCERTAINTY
Table 5 Results from stage 2 (variations in log kills)
Entries are comparative cure indices of the optimum schedule, the stage 1 rule,
and the alternating schedules (averaged).
Notation: + and - as in Table 4. S, cell types S and RB for drug A and S and
RA for drug B; R, cell types RA and RAB for drug A and RB and RAB for drug
B. For example, entry for A, S represents /f(A,S), which equals /f(A,RB) by the
assumption of no cross-resistance.
Kill parameters
RA S
-A -A -A -A +A
+A
+A
+A
-A
-A
-A
-A
+A
+A
+A
Index"
R-
IABI/IBAI6B6A+
+
IB3AI+
6B6A+
6A6B-
arise where the optimum strategy starts with, and continues for
an extended period, the drug which appears less effective.
The stage 1 simulations suggest that, among cell-kills, only
-tf(A,RB) and tf(B,RA) are important in choosing strategy. This
"stage 1 rule," when translated to the stage 2 setting, is; if
tf(A,S) > tf(B,S) then use 3A9B; if tf(A,S) < tf(B,S) then use
3B9A; otherwise use an alternating schedule. Table 5 presents
rule1.180.601.701.181.700.60optimum1.121.610.701.122.381.061.612.380.980.701.060.98
the results of using the optimum, the stage 1 rule, and an
alternating schedule. In each case, the stage 1 rule comes
substantially close to the optimum than the alternating sched
ules, indicating that partial information about cell kills on
subpopulations can be useful. However, in none of the asym
metrical cases does the optimum coincide with the stage 1 rule,
and the difference in cure index can be as high as 0.7.
Stage 3: Effects of Kinetic Parameters on Stage 2 Rules. In
stage 2, only the kill parameters had freedom to vary. If sym
metry is violated also for a single kinetic parameter, will the
"stage 2 rules" still perform well? By "stage 2 rules" we mean
Index* stage
Index'
IABI/IBAI+
+
9A3B+
IB3AI+
IA3BI+
+
9B3A+
+
+
IABI/IBAI+
+
9B3A+
6A6B+
IA3BI-1+
+
H9A3B+
the strategy of choosing the schedule which was optimal in the
+ IABI/IBAIalternating0.630.050.190.450.050.25-0.350.110.19-0.350.660.100.450.110.100.071
+ +BBBBBBBBBBBBBBBBS
stage 2 investigation, based on the cell kill asymmetry pattern.
" Average cure index of the two alternating schedules ¡Alt:and |BA|.
To answer this, the simulations of stage 2 were run again, but
* Cure index of schedule chosen according to the stage 1 rule: 3B9A if Ai(A,RB)
with altered values for the kinetics parameters, in every com
</T(B,RA), 3A9B if tf(B,RA) < Af(A,RB), alternating otherwise. If entry is empty
it agrees with the first column (alternating).
bination (16x2x4=
128 different models).
' Best cure index among the 16 schedules. If entry is empty it agrees with the
second column (stage 1 rule).
well occurs when the difference between sensitivity and resist
ance is very great for one drug and very little for the other. If
A"(A,S) and K(B,R) are relatively large and K(A.,R) and tf(B,S)
are small, then the schedule 9B3A beats alternating schedules
by more than 2 orders of magnitude in improved odds. The
early B doses are needed to keep down the cells resistant to A.
Drug A can wait, because B is controlling RB cells somewhat,
and A will be highly effective against them when it is finally
used. Note that A would appear to be more effective than B
according to single-drug clinical response. Thus situations can
The stage 2 rules were found to be fairly successful. In Fig.
44, it can be seen that the alternating schedules ¡AB;and ¡HA!
generally do very poorly compared to the true optimum. In
contrast, Fig. 4Äshows that the schedule which would be chosen
according to the stage 2 rules is fairly close to the true optimum
in the majority of cases. The three circled exceptions to the
stage 2 optima are runs in which D7"(RA) takes on its low value.
In 2 of these 3 runs, the stage 1 rule (9B3A) for low DT(RA)
performs well. However, in the fourth run with low OT^RA),
the worst drug rule does well, while the other poor performances
of the rule do not involve low D7"(RA). Thus the exceptions
also follow rules, but not as reliably as one would like.
In many of these simulations, one-drug schedules did sub-
Pr (no cells)"
RD '':
RD
;
•»i1.50-1
H
lJL•
Ì,
.,
\30
190
'19«J*B'-^___
.50-1
Fig. 3. Detailed demonstration of the worst drug
rule.
model and
three schedules
Fig.
2 areThe
presented
in detail.
A'(A.S) iscircled
largerin than
A(H.S), yet schedules with more doses of B, 3A9B
and |A3B| perform better. Prieure) = 0.007 for |AB|.
0.09 for 3A9B, and 0.21 for |A3B|.
'Ordinate, probability that there are no cells of
the specified cell type (7",total).
*Ordinate, average number of cells conditioned
on more than zero cells; for example, the curve for
RA at 33 months shows the "average" number of
cells among tumors that still have RA cells just
before the 33-month treatment. (The 7"curve may
.01-, RB
.90-1
30
33
36
39
42
month*
33
3639
33
36
42
month
(cond'
to".
dip below another, because each curve is conditional
on a different event.) The average used is roughly
the median: see Ref. 13.
AUADADADADAD
¡AB!
AAADDDDDDDDD
3A9B
39
ADD BAD DBA DDD
¡A3B!
3881
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42
TREATMENT
3.00
SEQUENCING,
ASYMMETRY,
+
failure to control sensitive cells. It also fails when the stronger
drug has an advantage on the cells resistant to it [e.g., if tf(A,S)
> K(B,S) and K(A,R) > K(B,R)]\ thus, even though log kills on
resistant cells are very small, they are important because the
target cells are important and few enough to be affected.
2. Cross-resistance greatly affects the magnitude of benefit
gained from using two-drug schedules instead of one-drug
schedules (high cross-resistance being unfavorable, of course).
This is no surprise. However, cross-resistance and collateral
sensitivity have a special impact on the selection of a two-drug
schedule. The study of many combinations of cell kill values
showed that tf(A,RB) and /f(B,RA) determined the pattern of
performance for the two-drug schedules almost to the exclusion
of /f(A,S) and AXB,S)as important factors. Therefore the prob
lem with cross-resistance for schedule choice is the danger of
being misled by crude measures when attempting to apply the
worst drug rule. If the equalities describing lack of crossresistance are true, then one can judge whether A or B is the
"worst drug" by indicators of A"(A,S) and tf(B,S), e.g., crude
2.00+
MAX
1 .00 +
ALT
3 .00+
2.00 +
MAX
I .00 +
2
O
"*2.2
4253'
52?X
0 . 00 +
+- - 1 . 00
B
•
—t0.
.00
AND UNCERTAINTY
2.00
3.00
RULE2
Fig. 4. Stage 3 cure indices. *, 1 obs; 2-9, 2-9 obs; +, more than 9 obs. A,
best cure index (MAX) versus average cure index of alternating schedules (ALT).
Mean (MAX - ALT) = 0.94; SD (MAX -ALT) = 0.63. B, best cure index (MAX)
versus cure index under stage 2 rules (RULE 2). Mean (MAX - RULE 2) = 0.14;
SD (MAX - RULE 2) = 0.29.
stantially better than alternating schedules. As would be ex
pected from stage 1, the models for which 12A performed well
had low values for A( \.si and AT(B,R)and a high value for K
(A,R). Recall that in stage 2, these values made 12A as good as
alternating schedules. In stage 3, the relative value of 12A was
enhanced greatly by a low A/A(S,RA) or a high A//?(S,RB), a
high DT(RA) or a low DT(RB), a high DR(RA) or a low
DR(RB). In each of these cases, either the relative difficulty or
the relative importance of eradicating RAs is decreased. These
are the same conditions under which the worst drug rule per
forms worse than usual. The resulting advantage of 12A over
alternating schedules is in the range 0.70 to 1.80 on the logit
scale, which is quite substantial.
Variables which were unimportant in stage 1 are now seen to
be important in some circumstances. For example, if A/A(S,RA)
is low, AT(A,S)must be high for 12A to be an acceptable strategy,
but if A/A(S,RB) is high instead, the value of tf(A,S) is irrele
vant. The effect of /f(A,S) is masked, because the cause of
failure is overgrowth by RB and RAB cells. This kind of
synergism appears throughout our experience with these
models, illustrating the hazards of simplifying assumptions and
suggesting that precise predictions of optimal treatment strat
egies are impossible given the imprecision of currently available
knowledge.
Further Departures from Symmetry. Several additional stages
of simulations were performed, involving simultaneous varia
tions of many parameters.
The results are summarized as follows.
1. The worst drug rule is valid over a wide range of parameter
choices, but it can fail badly, when treatment failure stems from
response rates or regression rates. Otherwise, simple analysis
of clinical results cannot reveal which is the worst drug.
3. In further stages, all parameters were allowed to vary
simultaneously, to get a stringent test of any potential treatment
rules. A vast set of models in this perverse realm of extreme
asymmetry was surveyed through systematic sampling. The
optimal schedules of the Stage 2 rules did not perform much
better than alternating. However, a less ambitious version of
the worst drug rule did perform well; the worst drug rule did
reliably predict the best dose mix (i.e., the total number of As
and Bs in the optimal schedule), but not the best time mix (the
sequence of the treatments). The optimum timing was affected
substantially by the kinetics.
DISCUSSION
This work reaffirms the observation of Goldie and Coldman
that in theory changes in treatment schedule could change the
outcome significantly. The advantage of using a two-drug sched
ule over one drug was enormous in most models discussed
herein. Among two-drug schedules, the advantage of alternating
schedules in symmetrical models was often substantial. The
extra advantage of administering therapies simultaneously in
stead of alternating was found to be small in the symmetrical
model.
The major new lesson is that optimal strategy can depart
dramatically from alternation as a consequence of drug asym
metry. The advantage of using the best schedule rather than an
alternating schedule is usually substantial, typically exceeding
the advantage, in the symmetrical model, of alternating (jABj,
say) instead of switching just once (6A6B, say). Also, the effects
of asymmetries in the drug sensitivity of subpopulations on
schedule performance are well summarized over a wide range
of models by the worst drug rule. In the worst cases, asymme
tries in kinetics affect the optimal timing of the treatments
drastically, but only rarely does the optimal proportion of each
drug change. In the best cases, when the asymmetry is minor
or else accurately measured, the optimal schedule could actually
be predictable beforehand.
How should these results be used to plan treatments and
protocols? The answer depends strongly on several distinct
kinds of uncertainty. There are interpretation problems, as well
as sampling and measurement errors, in assays for the param
eters of an individual patient's tumors. There is uncertainty in
estimating from clinical data the characteristic
kinetics of a
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TREATMENT
SEQUENCING,
ASYMMETRY,
tumor type and vulnerability to the commonly used drugs. A
more subtle and possibly more crucial type of uncertainty stems
from the fundamental variability of tumor parameters among
patients with the same tumor type. This is discussed next.
Pattern Homogeneity and Patient Variation
A critical question is whether cell kinetics and cell kill param
eters vary greatly among patients who are ordinarily regarded
for clinical purposes as having the same tumor type. Any
protocol strategy which aims at producing a single two-drug
treatment schedule for all patients with the "same" tumor type
will meet with limited success, unless the variation among their
tumors is small in some sense.
Suppose there are large differences among "similar" patients
(e.g., differences in mode of resistance, pharmacokinetics, or
modes of relaxed genetic control), leading to differences not
just in the values of the various parameters, but also in their
relative sizes, i.e., their pattern. Then the asymmetries are
qualitatively different from patient to patient, so the optimal
strategy is likely to be different for each patient. Changing the
sequencing and timing would benefit some at the expense of
others. Protocols testing changes in the order of delivery and
the total dose mix would bear little fruit.
However, suppose that these patients have the same tumor
parameters or that the departures from symmetry are qualita
tively the same for each patient. Then rules derived from tumor
models could be very useful. Optimum strategy would be
roughly the same for all patients in the diagnostic category.
We call this condition "pattern homogeneity," to emphasize
its importance. This is a milder condition than it might appear.
To be more precise, we are requiring that the optimal treatment
schedule choice would be roughly the same for all patients if
one could make the choice by measuring each patient's tumor
parameters exactly and computing the cure rate for a set of test
schedules. (Of course this is not requiring that such measure
ments actually be possible.) The results of this study suggest a
simpler formulation, that the "worst drug" should be the same
for all patients [e.g., either tf(A,RB) > A^RA) or vice versa,
uniformly for all patients], and the cell kinetics asymmetries
should not be too large.
In judging whether patient variability forecloses the hope that
pattern homogeneity exists among clinical tumors, one must
not be discouraged by the observed variability in the clinical
course of the targeted disease. Patients would appear to differ
even if they all had the same parameters, simply because the
random nature of mutation leads to different numbers of re
sistant cells. In addition, pattern homogeneity need only hold
within a subgroup of those patients with the best chance at cure,
and if the subgroup is large these patients would not have to be
identifiable for the new strategy to work well.
A treatment trial with animals is probably the most useful
initial approach to the question. If treatment outcome turns out
to depend strongly on schedule, the hope for pattern homoge
neity is reinforced. However, since experimental animals are
genetically homogeneous, pattern homogeneity is probably built
in, even for nonimplanted tumors. Therefore a positive result
would not necessarily extend to human patients. On the other
hand, a negative result would be conclusive. Animal trials to
date have focused on drug combinations and dose levels; there
seem to be no studies of scheduling in isolation from other
effects.
AND UNCERTAINTY
Three Protocol Strategies
1. A Shotgun Approach to Protocol Design. Suppose that a
diagnostic category is homogeneous enough to make a search
for an optimum schedule worth consideration. The strong the
oretical dependence of outcome on treatment schedule which
we have seen suggests the simple strategy of testing many
treatment schedules without attempting to predict the optimal
schedule. The high likelihood of parameter asymmetries would
offer a strong hope that standard drug combinations could be
measurably improved merely by changing the schedule. There
would even be a reasonable chance that effective strategies could
be found in tumors appearing incurable now. In that case,
protocols should test new schedules as well as new agents, and
the variety of schedules to be tested should be bold, including
especially treatment arms which give a major role to drugs
viewed as considerably less effective than the best available.
The success of this approach depends heavily on the presence
of pattern homogeneity and on the availability of increased
clinical trial resources.
2. Applying the Worst Drug Rule. Can one predict which
schedules might work best, using easily available data? The
worst drug rule, the concept of using the less effective of two
drugs first or for greater duration, is intriguing in this regard.
Its application requires a judgment as to which drug really is
the worst. One might approach this through clinical observa
tions of the ability of each drug to induce response or through
an appropriate cell kill assay.
The worst drug rule improves the cure rate by controlling the
cells resistant to the stronger drug. The stronger drug, by
controlling sensitive cells, is better at preventing newly resistant
cells from arising from sensitive cells during the treatment
period, but this effect appears usually to be of less value than
controlling the resistant cells already present.
This concept is generally contrary to several "common sense"
rules of cancer treatment, e.g.: "use the best drug(s) first; switch
to others when the first treatment fails"; "a drug which is poor
at inducing clinical response is not useful"; "first induce a
remission—worry about maintaining it later"; "when survival
is no different between two treatment arms in a clinical trial,
compare response rates to determine which will be the control
arm for further trials."
The worst drug rule suggests that the distinction between
induction and maintenance is artificial and counterproductive.
In fact, protocol design practice may unwittingly have plotted
out the slowest course to optimal treatment schedules. This
may help to explain the glacial progress of combination chem
otherapy for most tumor types, despite the many signs that the
mutation theory of drug resistance is correct. The novelty of
the worst drug rule means that there is unlikely to be any
clinical experience to date that can rule out its usefulness. Since
the current common practice is not even to alternate but rather
to use the opposite of the worst drug rule, the potential benefits
of such a change in strategy are if anything underestimated.
The usefulness of this strategy in practice would rest on
several conditions.
1. When the optimum schedule begins with the weaker drug,
the weaker drug would have to be at least strong enough to
limit S-cell-mediated morbidity temporarily. This problem may
be avoidable. Generally, the optimum schedule was found to
begin with the stronger drug but to switch very early. Even
when a schedule with the weaker drug delivered early performed
better, the extra advantage was rarely large. Since this schedule
was occasionally very poor, it may be wise to restrict attention
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TREATMENT SEQUENCING,
ASYMMETRY, AND UNCERTAINTY
to the schedule which begins with the stronger drug and
switches very early (possibly to switch back).
2. Even when morbidity from the tumor is not an immediate
problem, the treating physician must be psychologically pre
pared to walk a tightrope. The moral dilemma raised by switch
ing from a treatment that appears to be inducing a response or
treating with one drug when another would give more imme
diately satisfying results (although possibly at the cost of the
patient's long-term survival) is not to be underestimated.
3. There must be no evidence that the strategy might fail due
to cell kinetics asymmetries. We have seen that failure of the
worst drug rule may be caused by asymmetries in growth rates,
mutation rates, or death rates. If the growth rate for RA cells
is lower than for S-cells, then the RA cells are less of a problem,
so the control of S and RB cells becomes relatively more
important. If mutation rates to single resistance are lower than
mutation rates from single to double resistance, then the relative
importance of preventing new singly resistant cells by control
ling S-cells is increased. If death rates of singly resistant cells
are high, the effective mutation rate to double resistance is
raised (15), and the ability of small numbers of singly resistant
cells to survive is impaired; both effects tend to decrease the
value of the worst drug rule.
All three asymmetries are biologically plausible. Resistant
cells probably do grow slowly at first (16, 17). Mutant cells
probably do have higher mutation rates; this would be a prob
able consequence if the degree of accuracy in replication of the
genome is partly heritable, as several researchers have suggested
(4,18). Finally, mutant cells could be more vulnerable to normal
wear and tear.
Although extreme kinetics asymmetry can alter the optimal
strategy and cause the worst drug rule to perform poorly, some
schedule emphasizing the worst drug is usually found to be
nearly optimal. Therefore, if one has the luxury of testing a
variety of schedules, the worst drug rule can be a useful guide
to reducing the number of schedules requiring testing in the
shotgun protocol strategy.
4. Treatment failure in regimens using only the stronger
drug would have to be attributable to heritable resistance to this
drug. (Heritability in the loosest sense is sufficient, encompass
ing, for example, even physical location in the sanctuary of the
central nervous system.) The cells responsible for failure would
have to be substantially more responsive to the weaker drug.
5. The worst drug must be identifiable. Remember that the
crucial element is the log kill of each drug on the cells resistant
to the other. Judging which drug is the worst by response rates
to single agents, which reflect mostly the sensitive cells, is
potentially perilous when there is cross-resistance or collateral
sensitivity. Cross-resistance affects the size of the benefit to be
achieved through combination, but not the optimal schedule
itself, once A"(A,RB) and tf(B,RA) are known.
Pattern homogeneity would again be a prerequisite for suc
cess if the worst drug is chosen on a population basis, e.g., if
response results from clinical trials were used. This would not
be necessary if accurate assays for cell kill were available on an
individual basis. The work presented here therefore encourages
vigorous efforts to improve assays for cell kill, especially on
resistant subpopulations.
3. Using the Biology to Refine Strategy. Could one do even
better by using laboratory assays of each individual's tumor to
obtain estimates for the kinetics parameters and serial mea
surements of sizes of the subpopulations? If so, the need for
pattern homogeneity would evaporate. An intensive study of all
relevant information for a single well-chosen tumor type seems
to hold great potential. Nevertheless, the task would be sizeable.
Our work here makes it clear that plugging one plausible set of
parameter values into a model is insufficient; careful evaluation
of all the uncertainties is required. Other constraints on the
prospects for improving treatment with current assay technol
ogy include problems in defining and measuring the relevant
quantities, as the controversies surrounding the use of clonogenic assays illustrate (19). Finally, the population biology of
the tumor is immensely more complicated than any model;
aspects unaccounted for could easily sabotage attempts at a
priori treatment schedule choices. These difficulties dampen
hopes that kinetics and log-kills will soon be measured accu
rately enough to allow a priori choice of successful strategies.
On the other hand, the calculations in this paper are averages
over all randomly generated outcomes; laboratory techniques
which could accurately count in vivo resistant subpopulations
even when they are small would identify the outcome of the
random process of tumor growth and thereby confer an advan
tage even greater than perfect knowledge of the parameters.
Cautions
Several cautionary notes are in order.
1. The models we examined are still limited compared to the
actual range of tumor population behavior. The results we have
presented could depend on assumptions which were not re
moved, just as the value of alternating has been shown to change
when simplifying assumptions are removed. The discovery of
"pleiotropic resistance" (20-22) suggests including non-crossresistant and cross-resistant subpopulations in a single model.
(Should this phenomenon prove to have great clinical signifi
cance, the incorporation of countermeasures into standard
treatment regimens may gain from consideration of the points
raised in this paper, especially the worst drug rule.) The role of
gene amplification in drug resistance (17) suggests investigating
multiple-step models for resistance. Some other potentially
important modifications related to tumor heterogeneity include
metastasis (4, 18, 23), interactions with the immune system
(24, 25), cell dormancy (26), and the patterns of differentiation
and self-renewal in tumors (5, 27-29).
Very small subpopulations (stem cells which are drug-resist
ant metastatic progenitors) may be the desired targets; strategies
may succeed or fail based on the fate of subpopulations not
reflected by the clinical response of the entire tumor. The use
of the logit scale for comparing cure rates is intended partly to
make the results robust to omissions and misspecifications in
the modeling, but adequacy of this device has been checked
only in limited ways. The computer program used for our
simulations is flexible enough to handle each of these compli
cations by including whatever subpopulations are desired, but
as the model becomes complex, both computing time and the
difficulties of framing the proper questions grow.
2. Our results do not directly address how long to continue
treatment after induction of a complete clinical response or
how to choose drug combinations when several drugs are avail
able.
3. We considered cure rate only, but in the clinic cure rates
are often immeasurably small and may be unachievable. Then
survival may be a better criterion for judging schedules. We
have not dealt with the issue of total tumor burden and its
relationship to mortality and morbidity. Nor have toxicity
considerations been addressed.
4. The mapping of the concepts presented here into clinical
practice is not automatic. Each disease site has its own time
frame and its own body of laboratory knowledge; each drug
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TREATMENT
SEQUENCING,
ASYMMETRY,
menu has special constraints absent from our model, such as
toxicity and phase specificity. A specific application requires a
dedication to sophisticated cooperation between physicians who
understand the clinical constraints, researchers who understand
the site-specific biology, and statisticians who understand the
properties of models and data.
5. The concepts presented here might become obsolete by
virtue of new developments. The worst drug rule may become
moot if further investigation proves that tumors are so highly
variable that even the weak requirements for pattern homoge
neity fail. This would be useful to know, because it would
identify a fundamental limit on the success of multidrug regi
mens. In that case, restricting schedule protocols to alternating
schedules would be a safe and reasonable strategy, although
unlikely to do much better that other two-drug schedules.
On the other hand, the relevance of pattern homogeneity as
a requirement for progress may recede as techniques for assay
ing subpopulations improve. Simultaneously, better quality
data should make the tumor modeling tools more immediately
useful, since most forseeable new modalities seem likely to
continue to be limited by toxicity and by "mutation to resist
ance" in some sense.
ACKNOWLEDGMENTS
Many thanks go to S. Lakagos, M. Buyse, D. Amato, K. Propert,
and J. Cairns for careful readings of this paper. Thanks also to A.
Goldman, W. Peters, S. Bernal, M. Zelen, A. Tsiatis, H. Skipper, E.
Frei III, V. Ling, and J. Till for stimulating discussions and to M.
Pagano of the Health Sciences Computing Facility, Harvard School of
Public Health, who graciously provided computing resources.
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Treatment Sequencing, Asymmetry, and Uncertainty: Protocol
Strategies for Combination Chemotherapy
Roger S. Day
Cancer Res 1986;46:3876-3885.
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