In this paper, we have presented a combined
mathematical modeling and experimental approach to investigate the
effects of combination treatment strategies and schedules on the evolution
of acquired resistance in non-small cell lung cancer. Our results
suggest that optimally timed combination strategies may achieve dramatic
improvements in overall outcome over monotherapy with the same drugs
and concentration. In addition, we found that applying high doses
to achieve the fastest possible tumor reduction rate initially was
not always the best strategy in the long term, as this additionally
led to a maximal selective pressure, which was rapidly evaded by the
acquisition of resistance mutations.

The apparent strength of
targeted therapies such as erlotinib is the ability to strongly inhibit
populations harboring the specific molecular target, leading to a
dramatic tumor size reduction if these sensitive populations comprise
the majority of the initial population. However, such strategies may
lead to progression of disease (POD) due to the outgrowth of a resistant
subpopulation. Using our mathematical model to search the space of
schedules, we identified regions on the dose-combination map in which
theoretical elimination of the NSCLC cell population was possible.
We observed that, as a direct consequence of the growth rate response
curve, continuous paclitaxel schedules above 18 nM resulted in eventual
elimination of the tumor. However, information is not available about
whether a sustained plasma concentration of the drug at this level
can be tolerated in patients; thus we proceeded to investigate all
regions of the dose-combination map where elimination of the cell
population was possible. For example, we found that a sequential schedule
combining 6 μM erlotinib for 28–37% of the time with
15 nM paclitaxel for the remaining time was predicted to lead to an
eventual elimination of the tumor cells. At these doses, monotherapy
with erlotinib resulted in an initial tumor reduction followed by
POD, consistent with clinical observations, and monotherapy with paclitaxel
resulted in a lack of response.^{1,16,18} In contrast, the alternating strategy was predicted to lead to elimination
of the cancer cell population. When the erlotinib concentration was
increased beyond 6 μM, favorable schedules were found at even
lower paclitaxel concentrations and shorter treatment times. For treatment
schedules for which NSCLC cell elimination was not possible, we investigated
a wide range of scheduling strategies and identified both the optimal
outcome, defined as the maximal time until POD, and a schedule that
achieved this outcome. We found that the strategies that maximally
delayed POD were closer to the center of the dose-combination map
with paclitaxel doses between 10 and 15 nM and erlotinib at doses
between 1 and 10 μM approximately symmetrically pulsed (~50%
of time on each drug).

These results suggest that sequentially
pairing a targeted inhibitor with a cytotoxic drug inducing a weak
differential selection pressure on the TKI-sensitive and TKI-resistant
populations may lead to a better overall outcome. The inhibitory effect
on the resistant cell population by the cytotoxic agent was sufficient
to enable the design of an alternating pulsed strategy that controlled
and eventually eliminated both cell populations. Although the initial
rate of tumor reduction was not as dramatic using these sequential
combination strategies as was monotherapy of a molecularly targeted
drug, we predicted that sequential combination strategies led to slow
tumor elimination rather than POD due to resistance. Thus, the moral
of the story from the tortoise and the hare, “slow and steady
wins the race”, also seems to apply when designing treatment
strategies. We hypothesize that there are many existing cytotoxic
therapies that, at low or moderate doses, induce a slight inhibition
of the growth of cells with resistance to targeted therapies. At these
low doses, these drugs are good candidates for sequential combination
trials. We predict that sequential combination therapy will provide
a better, less costly alternative to the development of second generation
molecularly targeted inhibitors for the resistant cell populations,
which are themselves intrinsically vulnerable to additional resistance
mechanisms.

Supporting these findings, several human and mouse
trials of NSCLC suggested that sequential therapy using a cytotoxic
agent and either erlotinib or gefitinib was more effective than monotherapy
with either drug or with combination concurrent dosing.

^{26,27,29,42} It was postulated that doses or timing of such sequential therapy
would greatly influence the outcome. Our mathematical model predicted
that sequential therapy does indeed provide a better outcome than
either therapy alone at the same doses. The overall outcome was sensitive
to timing, dose, and initial ratio of sensitive to resistant cells,
and we were able to identify the correct balance of pulses to overcome
TKI resistance. We predicted, and validated experimentally, that the
initial ratio of resistant to sensitive cells influenced the overall
time until POD. Identification of noninvasive methods for monitoring
the molecular genotypes of tumors throughout a course of treatment
would provide valuable real time data that could be used to dynamically
update the model to help further guide treatment schedules. More specifically,
this information could be used to define the threshold of resistant
cells at which a particular treatment strategy would succeed or fail.
Several groups are working toward developing such platforms that would
routinely analyze circulating tumor cells or tumor DNA in plasma to
provide quantitative molecular characterization of tumors.(

19)

We recognize that, as with most models,
our framework has limitations. First, the two NSCLC lines used in
our biological model are not isogenic. However, they were chosen to
train our model for several reasons: (1) they carry known EGFR mutations
that are observed clinically and confer sensitivity and resistance,
respectively, to EGFR TKI therapies; (2) there exists a significant
differential growth rate between these two lines during treatment
with EGFR TKIs; and (3) we are interested in modeling the penetrance
of resistance and therefore are not considering the rate of new mutations
that would convert sensitive cells to resistant cells. Another limitation
is that the data used as input to the model was derived from an *in vitro* system. Thus, pharmacokinetic processes present *in vivo* such as absorption and elimination of the drug were
neglected as well as potential drug interactions. Furthermore, interactions
with endothelial, mesenchymal, and immune system cells were not considered
in our model. However, even with these idealized strategies and rates
measured *in vitro*, we were able to recapitulate key clinical findings.
As long as the relative relationships between growth rates *in vitro* are similar to those *in vivo*,
relative benefits of various scheduling paradigms can be evaluated
in our system, and our predictions will provide starting points for
preclinical and clinical evaluation of sequential combination therapies.
Thus, although the model has limitations in terms of precise clinical
predictions, two main conclusions can be drawn: the timing of drug
scheduling in combination therapies can have a striking impact on
the overall outcome of therapy, and mathematical modeling provides
a useful and efficient method to investigate and optimize over the
multidimensional space of scheduling strategies. These realizations
serve as a starting point for future investigations that will address
more complex scenarios arising *in vivo* as well as
additional resistance mechanisms and drugs. Our approach will also
be useful for investigating the dynamics of resistance against targeted
therapies for other tumor types.