Current cancer therapeutic strategies are not always maximally effective. Thus, our goal is to understand the biological basis of drug resistance and how these insights may improve therapy. Traditionally it is thought that drug-resistance results from genetic mutations and that reducing the population of proliferative tumor cells reduces the size of the target that can acquire resistance [52
]. In some cases, a higher dose of drug will kill drug-sensitive cells faster than the same drug at a lower concentration [45
]. Thus, it makes sense that “oncologists are usually trained to expect maximal benefit at the maximal dose” [54
]. The maximal dose administered is known as the maximum-tolerated dose (MTD). Doses exceeding the MTD would cause morbidity and mortality, also known as dose-limiting toxicity (DLT). The current strategy of maximum-tolerated dose means killing the cancer just before killing the patient, more is better, or “no pain no gain” [56
]. Many regimens of chemotherapy are, in fact, begun with cycles of maximum-tolerated-dose therapy, a strategy known as debulking or induction [55
Paradoxically, however, MTD is not always best. This has now been illustrated in a variety of cancers. In 2008, Seidman et al.
reported on a clinical trial of women with metastatic breast cancer [58
]. Women randomized to one trial arm received the drug paclitaxel at MTD. Doses were separated by rest periods of 3 weeks to allow for patient recovery, e.g. of bone marrow function. Women randomized to an alternative schedule received the same drug paclitaxel, but at a lower dose allowing more frequent administration, once every week, rather than once every three weeks. High-frequency, low-dose therapy has been called “maintenance” or “continuation” therapy and more recently “metronomic” therapy [57
]. While the survival rate after several years for the advanced condition in both arms of this study was low, overall survival in the cohort treated metronomically at 6 years was, remarkably, quadruple the overall survival in the cohort treated conventionally.
In another clinical trial, Klingebiel et al.
reported on the treatment of children with metastatic rhabdomyosarcoma, a cancer of muscle tissue [60
]. Both arms of this trial began with a period of conventional MTD therapy. Children in one study arm were then followed with 2 cycles of very high-dose chemotherapy with autologous stem cell support. Children in the other arm instead received follow-up with metronomic chemotherapy. In this study, the increase in overall survival was dramatic in both absolute and relative terms. Both survival curves plateaued after 3 years, but whereas the high-dose chemotherapy group displayed 15% overall survival, the group treated metronomically displayed 52% overall survival.
These results followed earlier hints of a beneficial role for metronomic scheduling including clinical responses during metronomic paclitaxel treatment in patients already pre-treated with taxanes [61
], an advantage in overall survival for children with acute lymphoblastic leukemia (ALL) treated with extended maintenance therapy [63
], and an advantage in overall survival for young adults with ALL treated according to metronomic pediatric schedules, rather than adult schedules [65
]. For these, and additional applications, high frequency dosing has been reported or postulated to be effective (Supplemental Table 1
If metronomic therapy has provided clinical benefit and sometimes advantage over conventionally scheduled MTD therapy, why has metronomic therapy not been more broadly adopted as a standard of care? Klement and Kamen have commented that there is “a random and very erratic manner in which doses and frequencies are chosen” [66
]. Even though the clinical trials for metastatic breast cancer and metastatic rhabdomyosarcoma described above both demonstrated a survival advantage for metronomic therapy, the advantage in one case was dramatic while in the other minimal. The purpose of designing the metronomogram is to try to identify variables that will maximize effectiveness of metronomic therapy.
5.1. To deplete the therapeutically targeted population, the generation of drug-sensitivity should be faster than net population expansion
Based on the insights we have presented in this paper, we hypothesize that metronomic therapy can be effective when drug-resistant cells convert to a drug-sensitive phenotype. Persistent, frequent application of low-doses of drug then depletes the emerging drug-sensitive cells before they have time to proliferate or become repatriated to the drug-resistant subpopulation. The overall dynamics of the target population are a combined result of drug kill and expansion during periods of drug-free recovery. In order for frequent low doses of therapy to be effective, the rate of restoring full homeostatic heterogeneity in drug-sensitivity must be greater than the rate of population expansion. This requirement relates time-scales for the generation of heterogeneity in drug sensitivity to time-scales for the overall expansion of cell population number. To compare these processes quantitatively, we next introduce a graphical device called a metronomogram that expresses the efficacy of drug administration frequencies applied to a target cell population undergoing phenotypic interconversion.
To relate the cellular processes essential for developing this tool, we provide a toy model (). In this timeline, cell populations are cyclically killed. The surviving populations repeatedly regenerate heterogeneous mixtures of drug-sensitive and drug-resistant individuals during time intervals between drug administration. For clarity, we take advantage of the simplifying assumptions, made elsewhere, that drug-resistant cells display no drug response and that drug-sensitive cells die immediately upon exposure to drug [48
]. Thus, instantaneous drug-kill occurs periodically at time intervals of duration Δt
. The total number of cells at any time is N
), i.e. the total population of both drug-sensitive and drug-resistant cells immediately following the drug-kill event at time t
= 0 is N
) while N
) and N
), respectively, denote the populations immediately preceding and following drug-kill at time t
Figure 4 Using a metronomogram to graph the dynamics of phenotypic interconversion and expansion in a heterogeneous population of cells. (a) Dynamics of a mixture of drug-sensitive and drug-resistant subpopulations with rest periods of duration Δt between (more ...)
We seek to reduce the tumor population with each cycle so that the population surviving after drug kill at time t = Δt is smaller than the population surviving the previous event of drug kill at time t = 0.
Adding to both sides the population immediately preceding drug kill at time t = Δt, we determine
that the increase in the population during a kill-free interval of duration Δt
(left-hand side of the equation) must be less than the decrease in population associated with drug-kill at the end of the interval (right-hand side of the equation). Defining the kill fraction, or effective drug-sensitized fraction fS
), at time t
as the number of cells killed divided by the number of cells exposed to drug
and defining the population expansion fraction fP
) as the net number of cells added to a population during a drug-free interval divided by the number of cells at the end of the interval
we rewrite (4
) concisely as the requirement
that the kill fraction exceed the population expansion fraction. Following a round of drug-kill, the regeneration of the drug-sensitive subpopulation must occur, in this sense, faster than the generation of overall cell population number. If therapy continues until the total cancer cell population N
) falls below unity, the expected number of drug-resistant cells in the population will also fall below one, and by chance the entire cell population confronting the next dose of drug could be drug-sensitive. Drug-kill then eliminates the last residue of the target cell population. The duration of the total course of therapy is denoted TCD
is a condition for evaluating the efficacy of a particular dosing period Δt
. To compare different dosing frequencies, we express this condition graphically. The timeline in describes cellular population dynamics for a particular dosing time interval Δt
, corresponding to a particular pair of values fP
) and fS
), indicated by a single circle in . It is important to remember that both the drug-sensitized fraction fS
) and the fraction fP
) by which the population expands in (7
) can vary as functions of the drug-free interval Δt
. We can repeat the experiment in using the same kinds of cells and drug dose per administration, but exploring other values of Δt
. This produces additional values of fP
) and fS
) that trace out, for example, the solid curve in . The example curves in illustrate solutions to the Markov model in (1
) and (2
) with rate coefficients for population expansion and phenotypic interconversion chosen to have similar magnitude (see accompanying paper). We call an fS
plot a “metronomics nomogram” or “metronomogram.” The dynamics of a population of interconverting drug-sensitive and drug-resistant cells is a combined result of population reduction during periodic drug-kill and population expansion during intervals of drug-free rest. If restoration of homeostatic heterogeneity in drug-sensitivity is faster than population expansion, the population shrinks over time. This corresponds to positions on the metronomogram above the diagonal line fS
, which satisfy condition (7
). In contrast, the generation of drug-sensitivity is relatively slow in the region below the diagonal. Here, the population expands long term. The diagonal fS
distinguishes dosing frequencies that can reduce tumor size (above the diagonal) from dosing frequencies that allow tumor expansion (below the diagonal).
5.3. Choice of drug administration schedule
We now provide an example to illustrate how the metronomogram we have described can be used to choose dosing frequencies (). Consider the situation where we measure the cancer cell population at three times in : right before and right after one of the pulses of drug kill, as well as once immediately preceding the subsequent pulse of drug kill. Then the values of fS and fP can be determined and plotted in , e.g. circle 1.
Clinical uses of metronomograms. During therapy, metronomograms could facilitate (a) dosing frequency selection and (b) recommending changes in dosing strategy in response to changes in efficacy of therapy over time.
Because circle 1 turns out to fall below the fS = fP diagonal, another drug administration frequency needs to be found. Extrapolating along horizontal dashed line 2 suggests that a shorter drug-free rest interval Δt might bring us to circle 3, located within the burden-reducing region where fS > fP. As a first approximation we assume that the net replication rate coefficients (proliferation less clearance) of the drug-sensitive and drug-resistant populations are equal. If this approximation is accurate, then an attempt to realize circle 3 will succeed in achieving a data point with the expected fP, corresponding here to ~1.1 population doublings. We might obtain circle 3 itself, or another outcome, such as circle 4. Circle 4 has the expected horizontal position of circle 3, but the actual kill fraction achieved differs, i.e. circle 4 falls vertically below 3. Here circle 4 lies in the region below the fS = fP diagonal, representing dosing frequencies that fail to sustain reduction of tumor burden.
However, such a therapeutic “failure” is actually an opportunity for improving drug administration schedule. The gradual approach toward homeostatic heterogeneity we have described manifests as simple, smooth solutions to the Markov model in (1
) and (2
), which are plotted in . This qualitative simplicity makes it easy to use circle 1 and 4 to estimate visually the shape of the blue curve labeled 5 in . This curve then indicates a range of interdose rest periods that might provide points in the therapeutic region above the fS
diagonal. Upon treatment at a fast frequency, we might achieve circle 6 as expected. If toxicity proves intolerable at this schedule, the blue curve we have established can help identify slower dosing frequencies that nevertheless remain in the region fS
. For example, the more relaxed schedule represented by circle 7 might be tolerable long-term. Information obtained from “productive failures” would provide data to optimize dosing schedules for each individual.
Proceeding with this schedule, we would continue to monitor tumor burden frequently to look for changes in the rate coefficients in (1
) and (2
) reflecting long-term changes in tumor biology. Despite maintaining a fixed interval Δt
, we might find an increase in kill fraction as in , circle 8. We might drift toward the right on the previously established blue curve, reaching circle 9. Alternatively, we might fall to another curve containing circle 10. The positions of the curves on the metronomogram suggested by these additional data would help us identify modifications to dosing frequency. If we obtained circle 8, a larger range of dosing frequencies might prove capable of reducing tumor burden with less toxicity. Circle 9 would be consistent with the sharing of molecular mechanisms in phenotypic interconversion and proliferation. In this case, cytostatic agents (which would minimize rR
) might move us left, back to the original position above the fS
diagonal. We would address circle 10 by investigating ways to hasten the return to homeostatic heterogeneity (maximize cR
) in drug-sensitivity in a proliferation-independent, rather than a proliferation-dependent fashion, as discussed in the accompanying paper.
To explain , we focused on the simplifying situation where the fitnesses of the drug-sensitive and drug-resistant cells were equal during drug-free rest periods. This allowed the metronomogram to be used literally as a nomogram. However, this need not be the case. We could have obtained a different result for circle 4 in , horizontally displaced from its illustrated position. If circle 4 had shifted to the right, this would suggest that the fitness of the drug-resistant cells, rR
, is greater than the fitness of the drug-sensitive cells, rS
. A shift to the left would suggest the opposite. This result would have forced us to consider unequal fitnesses for different phenotypic compartments. Measurements of tumor cell population at various times during a drug-free interval would then be important for constraining the parameters in (1
) and (2
) necessary for calculating the rest periods Δt
capable of reducing tumor burden. Even in these cases, the metronomogram continues to provide a powerful graphical tool for understanding why some ranges of dosing schedules are appropriate for a tumor’s immediate biology, and for proposing revisions to dosing frequency in response to changes in tumor biology during extended therapy.
In the supplemental materials
, we discuss how this simplified analysis illustrates a surprising feature. For very high dosing frequencies, the marginal improvement in total course duration becomes minimal. This effect of “diminishing returns” is related to the finite time required for a purified drug-resistant population to generate drug-sensitivity. It may not be worthwhile to increase the dosing frequency arbitrarily at all cost. Even though this simple analysis does not explicitly describe toxicity, it nevertheless contrasts with the idea of “more is better” underlying MTD therapy.
5.4. Calculating total course duration
In addition to choosing appropriate dosing frequencies as we have just described, it is necessary to choose a sufficient total course duration. Therapy concluded prematurely can fail to sustain durable response despite initially successful reduction in tumor burden. The metronomogram can be used to estimate the necessary total course duration as a function of dosing interval Δt. The position of a point (fS(Δt), fP(Δt)) encodes the fold-reduction in tumor size that occurs over a dosing cycle of duration Δt
is the total course duration that must elapse before a tumor starting with NIM
cells at the i
nitiation of m
etronomic therapy is f
inally reduced to a size NF
. The number of orders of magnitude of fold-reduction in tumor cell population sought during the total course of therapy
divided by the number of orders of magnitude of fold-reduction in tumor cell population during each dosing cycle
gives a deterministic estimate of the number of dosing cycles required. Multiplied by the duration Δt
of each cycle, this ratio provides the total course duration in units of time.
To what size NF
do we seek to reduce the tumor burden? By the time we succeed in reducing the tumor cell population to just less than a single cell in the deterministic model, NF
< 1, we expect the cell population in the corresponding stochastic dynamics to teeter at the brink of stochastic extinction, if the population has not already become extinct. However, even choices of NF
> 1 may provide durable clinical response owing to interactions with the tissue microenvironment. For example, angiogenic and immunological barriers can prevent the expansion of microscopic metastatic colonies for years, e.g. tumor dormancy [67
In the supplemental information
, we discuss how in some cases the total course duration may be of similar order to the length of time the target cell population initially expanded preceding diagnosis and treatment. For some cancers this corresponds to a time-scale of years [68
In sections 5.3 and 5.4, we sought to identify variables to compare when choosing dosing frequencies and duration. It is important to compare timescales for the generation of heterogeneity with timescales for population expansion in order to evaluate whether the rate of the generation of heterogeneity is faster than the rate of the expansion of the population, as necessary for effective therapy.
Our perspective has been that therapy should be timed to kill drug-sensitized cells before they have time for proliferation and back-conversion to the drug-resistant phenotype. Because phenotypic interconversion is stochastic, drug-sensitive cells can emerge from the drug-resistant population at a variety of times. To kill all of these cells soon after they gain drug-sensitivity, drug doses must be applied frequently. To manage toxicity while sustaining high dosing frequencies over long course durations, we use less than the maximum-tolerated dose. The modeling we have used and its analysis using the metronomogram provide one of the simplest ways we can develop this understanding quantitatively. Understanding these ideas at this basic level is important for being able to recognize the same interplay (between the kinetics of the generation of heterogeneity, the kinetics of population expansion, and therapeutic timing) when it appears in more sophisticated models, e.g. accounting for concentration-dependent drug response and cell-cycle specificity [51
] or describing interconversion among continua of drug-sensitive and drug-resistant phenotypes [48