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Silva and Gatenby Biology Direct 2010, 5:25
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Open Access
RESEARCH
BioMed Central
© 2010 Silva and Gatenby; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Com-
mons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and
Research
A theoretical quantitative model for evolution of
cancer chemotherapy resistance
Ariosto S Silva and Robert A Gatenby*
Abstract
Background: Disseminated cancer remains a nearly uniformly fatal disease. While a number of effective
chemotherapies are available, tumors inevitably evolve resistance to these drugs ultimately resulting in treatment
failure and cancer progression. Causes for chemotherapy failure in cancer treatment reside in multiple levels: poor
vascularization, hypoxia, intratumoral high interstitial fluid pressure, and phenotypic resistance to drug-induced toxicity
through upregulated xenobiotic metabolism or DNA repair mechanisms and silencing of apoptotic pathways. We
propose that in order to understand the evolutionary dynamics that allow tumors to develop chemoresistance, a
comprehensive quantitative model must be used to descr ibe the interactions of cell resistance mechanisms and tumor
microenvironment during chemotherapy.
Ultimately, the purpose of this model is to identify the best strategies to treat different types of tumor (tumor
microenvironment, genetic/phenotypic tumor heterogeneity, tumor growth rate, etc.). We predict that the most
promising strategies are those that are both cytotoxic and apply a selective pressure for a phenotype that is less fit
than that of the original cancer population. This strategy, known as double bind, is different from the selection
process imposed by standard chemotherapy, which tends to produce a resistant population that simply upregulates
xenobiotic metabolism. In order to achieve this goal we propose to simulate different tumor progression and therapy
strategies (chemotherapy and glucose restriction) targeting stabilization of tumor size and minimization of
chemoresistance.
Results: This work confirms the prediction of previous mathematical models and simulations that suggested that
administration of chemotherapy with the goal of tumor stabilization instead of eradication would yield better results
(longer subject survival) than the use of maximum tolerated doses. Our simulations also indicate that the simultaneous
administration of chemotherapy and 2-deoxy-glucose does not optimize treatment outcome because when
simultaneously administered these drugs are antagonists. The best results were obtained when 2-deoxy-glucose was
followed by chemotherapy in two separate doses.
Conclusions: These results suggest that the maximum potential of a combined therapy may depend on how each of
the drugs modifies the evolutionary landscape and that a rational use of these properties may prevent or at least delay
relapse.
Reviewers: This article was reviewed by Dr Marek Kimmel and Dr Mark Little.
Background
Disseminated cancer remains a nearly uniformly fatal dis-
ease. While a number of initially effective chemothera-
pies are available, tumors inevitably develop resistance to
these drugs ultimately resulting in treatment failure and
cancer progression. Causes for chemotherapy failure in
cancer treatment reside in multiple levels: poor vascular-
ization, hypoxia, intratumoral high interstitial fluid pres-
sure, and phenotypic resistance to drug-induced toxicity
through up regulated xenobiotic metabolism or DNA
repair mechanisms and silencing of apoptotic pathways
[1-5].
Solid tumors may present both phenotypic and envi-
ronmental therapy resistance. Phenotypic resistance is
due to increased cell survival mechanisms, environmen-
* Correspondence: robert.gatenby@moffitt.org
1 Department of Radiology and Integrative Mathematical Oncology, Moffitt
Cancer Center, Tampa, FL, USA
Full list of author information is available at the end of the article
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tal resistance consists in reduced drug efficiency by
tumor microenvironmental conditions. Examples of envi-
ronmental resistance in solid tumors are hypoxia -which
reduces efficiency of radiotherapy-, slow diffusion of
drugs from blood into avascular regions of tumors and
pHe induced quiescence [6].
Clinical tumors are rarely detected before they reach a
size of 1 cubic centimeter so that even a minimum tumor
burden will contain around 109 cells [7]. In view of the
intrinsic genetic instability that is characteristically
observed in cancer phenotypes, a billion cells will form a
phenotypically and genotypically heterogeneous popula-
tion which may harbor small populations of cells which
are already chemoresistant. In other words, phenotypes
with at least some degree of resistance to therapy are
likely to be present even prior to its administration.
Frequently, the initial doses of chemotherapy eradicate
a significant fraction of the tumor population. However,
most tumors typically become resistant over time result-
ing in repopulation of the original tumor site and devel-
opment of other metastases [8].
Unless a cytotoxic therapy eradicates all cancer cells, its
application to a tumor population also produces evolu-
tionary selection forces that will select for the individuals
that are adapted to the therapy and, thus, fittest to these
conditions. In fact, this mechanism has been used to pro-
duce many chemoresistant cells lines [9-11].
A fundamental principle of chemotherapy is to use
drugs that are more toxic to tumor cells than to healthy
cells, the preferred target being replication mechanisms,
as many tumors replicate faster than the host tissue
(except for fast replicating tissue such as epithelium).
Unfortunately, tumors are not homogenously prolifera-
tive. Typically, only its outer rim is composed of replicat-
ing cells, while much of its mass consists of cells in
quiescent or even dying states [12].
Thus, the cells on the outer rim of the tumor are the fit-
test extant phenotype in the tumor in absence of treat-
ment. They are also the most readily targeted by
chemotherapy due to their proximity to vascularization
and their fast growth. This region of the tumor is thus
environmentally sensitive and is mostly composed of
phenotypically sensitive cells, even though some pheno-
typically resistant cells may be present.
The inner regions of a solid avascular tumor are often
hypoxic and acidic due to anaerobic glucose metabolism,
what leads to quiescence and increased chemoresistance.
The increasing distance from vascularization reduces the
concentration of drug in these regions of tumor, confer-
ring a second factor of environmental resistance [13,14].
It has been proposed that chemoresistant cells are typi-
cally, in the absence of therapy, less fit that chemosensi-
tive cells. This is due to the phenotypic cost of resistance.
That is, the resistance to cytotoxicity requires expendi-
ture of resources and energy to upregulate xenobiotic
metabolism or DNA repair pathways. This energy is,
thus, not available for proliferation resulting in a reduc-
tion in the fitness. Assuming that phenotypically resistant
cells are less fit and, therefore, less proliferative than their
chemo sensitive counterparts [15,16] -in this model we
consider chemotherapy targeting mainly proliferative
cells- it is reasonable to consider that most of phenotypi-
cally chemoresistant cells will be trapped inside the
tumor volume as the latter grows.
Consequently we propose that the inner region of a
tumor confers environmental resistance to cells there
residing due to reduced blood flow, hypoxia, and acidosis
and that it also harbors most of the phenotypically resis-
tant cells as well, making this area a candidate for origin
of tumor regrowth and source of chemo resistance.
Evolutionary double bind and cancer therapy
In this work we describe and model solid tumors as a
phenotypically heterogeneous cell population distributed
in an environmentally heterogeneous microenvironment
which supports proliferation when possible but also
rewards quiescence when needed.
Such a plastic evolutionary landscape shows similarities
to the control of invasive pests with chemicals: at first,
chemicals are capable of eradicating the pests almost
entirely, but after some generations, an entire population
of resistant individuals arises.
Centuries of experience in dealing with damaging
exotic species have demonstrated that best strategy is to
introduce into the ecosystem a predator, pathogen, or
parasitoid that targets the foreign population but not the
native species. These strategies work best when the inva-
sive species can adapt to the biological attack through
complex and costly phenotypic changes that substantially
reduce its fitness. This state, in which the pest faces either
attack by an enemy or costly adaptation that reduces its
ability to proliferate, has been termed an evolutionary
double bind [17].
The idea of changing cancer from a 'must cure' disease
to a chronic disease has been recently proposed by this
group [18] through computer and mathematical models
and tested in animal models showing promising results.
One potentially valuable strategy to achieve this involves
reducing the fitness of the tumor without actually causing
cytotoxicity. This could potentially stabilize the cancer
population without producing strong selective forces for
resistant phenotypes.
One specific form of this therapeutic strategy might
take advantage of the glycolytic metabolism commonly
observed in human cancers. Studies reporting the use of
glucose competitors such as 2-deoxyglucose to treat solid
tumors have been published for close to three decades
[19,20] and rely on the knowledge that the cells in tumor
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center regions are highly dependent on anaerobic glucose
metabolism due to hypoxia reigning in this region.
The rationale behind this strategy is that the same
hypoxic and acidic environment that confers environ-
mental chemoresistance to these cells also forces them to
increase their glycolytic metabolism to compensate for
the loss of energy due to reduced respiration described as
Pasteur Effect when this modification is transient and
Warburg Effect when it becomes independent of oxygen
availability [21]. These cells are more sensitive to glucose
restriction than healthy tissue or tumor cells closer to
tumor-host interface.
We propose that the increased robustness to chemo-
therapy provided by tumor hypoxic core also confers fra-
gility against glucose restriction and that this trade-off
can be instrumental to a double-bind strategy for treat-
ment.
Double bind applied to solid tumor treatment
In this work we propose to approach solid tumors che-
motherapy using an evolutionary double bind strategy
that weakens cells in both tumor rim and center, forcing
sensitive and resistant cells to compete with each other
leading to tumor stabilization, in worst case, or eradica-
tion, in best case scenario.
This approach assumes that the resistant phenotypes
are rare in population [22] and that resistance has a cost
leading to slower proliferation in free growth conditions
what eventually leads to phenotypically resistant cells
trapped in the inner regions of tumor while sensitive pro-
liferative cells are located in outer rim.
The strategy adopted in our model consists in targeting
the resistant cells with glucose competitors and lead them
to energy depletion and consequent cell death. The tumor
is kept stable with low doses (1 μM*h corresponding to
IC99 of sensitive cells or IC50 of resistant subpopulation)
in order to prolong patient survival until the resistant
population has been eradicated by competition with sen-
sitive ones.
We also evaluate the efficiency of different combina-
tions of chemotherapy and glucose competitor and the
effect of half-life of these drugs in system.
This is the first work, to our knowledge, that proposes
to identify the ideal conditions of combination of glucose
restriction and chemotherapy together with an analysis of
the progress of tumor resistance (phenotypically and
environmentally).
Methods
The model used in this work consists of a bidimensional
fixed-lattice cellular automata representation of a tumor
mass which was allowed to grow until it reached the size
of approximately 5,000 cells before treatment was started
(each cell represented as a 25 μm-side square), corre-
sponding to a total tumor diameter of 2 mm.
Each cell in the tumor had its intrinsic phenotypic
resistance and proliferative capacity which, according to
this model's assumption, are inversely correlated. In this
study, the following scenario was tested: sensitive popula-
tion with proliferative potential of 1 and IC50 of 10 nM*h
and resistant population with proliferative potential of
0.05 and IC50 of 1 μM*h.
A proliferative potential equal to 1 means that at every
generation, if a cell has enough energy and space to repli-
cate, it will do so. If a cell has a proliferative potential of
0.5, it can only replicate as fast as once every two genera-
tions of simulation and a cell with 0.05 will proliferate
once every 20 generations.
In this model, the interval between two treatment ses-
sions was arbitrarily defined as 5 generations, which in
human tumor cells would correspond to 1 month.
Diffusion, cell cycle, tumor growth
The proposed tumor is avascular and its growth is limited
by its cell's proliferative rate and energy availability, which
is a function of glucose and oxygen in interstitial fluid.
Oxygen, glucose and pH buffers diffuse through the
entire simulated volume isotropically and their concen-
trations are considered constant in blood vessels which
are the boundary conditions of this model [23]. The con-
centration of species in blood serum and diffusion con-
stants are listed in table 1[24-29].
In order to represent the gradient of nutrients from vas-
cularized tissue to tumor we represented the external sur-
face around tumor as vascularized tissue. This tissue does
not have fixed values for glucose, oxygen and other spe-
cies as blood vessels.
As tumor grows, it replaces this vascularized tissue.
Once therapy is applied to the system, the tumor recedes
but it loses contact with the vascularized tissue, until it
re-grows and gets in contact with it again.
The drug levels in blood were simplified to a step func-
tion where the concentration of drug is kept constant
during treatment and then the drug is completely
removed from system. The real behavior of drug concen-
tration in blood is an initial spike followed by an expo-
nential decay [30,31] which we approach by the equation
1:
Where T1/2 is the half life of drug in blood and t is time.
Integrating this function to infinity we obtain the AUC
(Area under curve of concentration versus time). So, the
width of a step function with the same AUC and bolus
intensity is given by:
()[ ] ()//
112
Rx(t) e t*ln2 T
=−
(2) T /ln2
1/2
τ=
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In this work, simulations of treatment considered a
duration of 5,000 s for drug exposure step function, cor-
responding to an AUC with exponential decay with half
life of approximately one hour, which is close to drugs
such as carboplatin (1-2 h).
Chemoresistance
Chemotherapy was represented as a hypothetical drug
with diffusion coefficient similar to glucose a half-life in
the order of one hour and having fast replicating cells as
target.
Drug resistance was implemented as a generalization of
the definition of IC50, which represents the drug concen-
tration and exposition time for which half the original
population survives. In this model we consider that the
probability of one specific cell to survive is given by the
dose response curve, meaning that if the extracellular
drug concentration for a given cell is IC50 the cell has a
chance of 50% of survival.
Each simulation step could be either (a) tumor free
growth or (b) therapy. During tumor free growth, no drug
is present and tumor cells are allowed to proliferate
restricted only by its energetic capacity, pH induced qui-
escence, proliferation rate and free space to grow.
During therapy, however, cells are subject to death
induced by chemotherapy as previously described, suffer
competition of its glucose transporters by 2-deoxy-glu-
cose and thus have their energetic capacity reduced
potentially leading to death by energy depletion or may
suffer low pHe induced death.
Metabolism, replication and cell death
In this model cell metabolism consisted in converting
glucose and oxygen into ATP, carbon dioxide and lactic
acid. Glucose and oxygen uptakes were proportional to
extracellular concentration [28].
In normoxic conditions, each molecule of glucose com-
bined with six molecules of oxygen to produce 36 mole-
cules of ATP and 6 molecules of CO2. In hypoxic
conditions the excess of glucose was metabolized anaero-
bically producing two molecules of ATP and two of lactic
acid.
Lactic acid (pKa = 3.86) was considered to dissociate
completely under physiological pH (~7.4). The extracel-
lular pH (pHe) in blood was fixed at 7.2 as was serum glu-
cose (5 mM) and oxygen (0.15 mM corresponding to 100
mmHg pO2) concentrations.
Low pHe could induce cell quiescence (pHe < pHQ, pH
quiescence threshold) or cell death (pHe < pHD, pH death
threshold) [32-34]. The values of pHQ and pHD are ran-
domly distributed among the cell population according to
a uniform distribution within the following intervals: 6.0
< pHD < 6.8 and 6.4 < pHQ < 7.1.
Cell proliferation rate was connected to its metabolic
capacity [29]. If ATP production was below a minimum
threshold (0.85 uM/s), the cell would die. Otherwise, if
ATP production was above this minimum threshold the
cell would have an increasing probability of replication,
reaching 100% at a maximum threshold (8.5 uM/s).
As previously described, cells with higher proliferative
phenotype have shorter cell cycle but will only replicate if
the energy requirements are met.
Simulations
In all simulations, the model was created initially with
cells with two subpopulations: one resistant and with low
proliferation rate and a second chemosensitive and highly
proliferative.
The tumor disc was allowed to grow until 5,000 cells
(corresponding to 270,000 cells in a spherical volume)
before the following scenarios were tested: (a) no therapy,
Table 1: Constants and values used in model
Constant Value Reference
H+ diffusion coefficient 1.08 × 10-5 cm2/s [24]
O2 diffusion coefficient 1.46 × 10-5 cm2/s [25]
Glucose diffusion coefficient 5 × 10-6 cm2/s [26]
O2 uptake rate 9.41 × 10-2 × [O2]/s [27]
Glucose uptake rate 5 × 10-5 × [Glc]/s [24]
Quiescence pH threshold for normal cells 7.1 [28]
Death pH threshold for normal cells 6.8 [28]
Mutation rate for tumor cells 0.1% [28]
ATP production for 100% probability of cell
duplication.
8.6 × 10-6 M/s [29]
Minimum ATP production for cell survival. 8.6 × 10-7 M/s [29]
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(b) high dose chemotherapy aiming tumor eradication,
(c) 2-deoxy-glucose only therapy, (d) simultaneous
administration of low dose chemotherapy with 2-deoxy-
glucose and (e) alternated administrations of chemother-
apy and glucose competitor.
Double Bind and MTD
High dose chemotherapy (maximum tolerated dose,
MTD) concentration was 5-fold the IC50 of the chemore-
sistant population (7 μM*h). And was allowed to perme-
ate the system for 5,000 s (1 h23 m). This number of
simulation steps was chosen arbitrarily in order to pro-
vide enough time for drug to diffuse to tumor center but
also short enough to comply with the half life of known
drugs.
This combination of diffusion coefficient and drug
exposure time leads to the gradients in figure 1, where the
concentration of drug in center of tumor is 80% of the one
in blood.
For the Double Bind therapy, the minimum amount of
drug was used to keep tumor size stable. The dose applied
was 20% of MTD, or 1.4 μM*h. After administering the
dose, tumor was allowed to re-grow and higher doses
were used if tumor size was bigger than original dimen-
sions or lower doses if the tumor was smaller then the
original. If no growth was detected no therapy was
administered.
2-Deoxy-Glucose
2-deoxy-glucose therapy was tested for concentrations in
the order of magnitude of glucose in blood serum. The
maximum dose, in the model, was determined to be 3
mM. Higher concentrations induced death of cells in
healthy tissue further than 150 μm from blood vessels
due to energy depletion. In animal experiments chronic
doses (diet) of 150 μM/l and bolus (i.p. injection) of 12
mM [35,36] have been used.
In all simulations in this work, 2-deoxy-glucose was
allowed to diffuse in model for 5,000 s and the concentra-
tion in center of tumor was around 85% of the concentra-
tion in tumor rim (2.5 mM and 3 mM respectively).
Combination of strategies
We also tested if the alternative combinations of chemo-
therapy and 2-deoxy-glucose would have better results
than the simultaneous administration. The combination
was tested as (a) chemotherapy and 2-deoxy-glucose
administered in tandem, (b) 2-deoxy-glucose administra-
tion followed by chemotherapy and (c) chemotherapy fol-
lowed by 2-deoxy-glucose.
Results
Previous in vitro and in vivo works [36] have shown that
exposing tumor cells to 2-deoxy-glucose concentrations
on the order of those considered in this study combined
with chemotherapy could provide better results than che-
Figure 1 Gradient of drug concentration for both chemotherapy and glucose competitor. Gradient of drug concentration for both chemother-
apy and glucose competitor. The longer the system is exposed to drugs, the higher the concentration in tumor center and more efficient the treat-
ment.
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motherapy alone. These studies, however, used simulta-
neous administration of these drugs and did not consider
sequential application.
Recent theoretical/in vitro models [37] have pointed
the diffusion of chemotherapeutics through tumor
towards its center as one of the limitations of success in
solid tumor treatment. According this model the concen-
tration of drug can decrease 50% at a distance of three
cells from tumor-host interface in steady-state and up to
80% across the same distance in a transient of 2 hours.
The gradients of drug concentration in our models
reached steady state after 5,000 s (~1.5 h) and corre-
sponded to a decrease in concentration of the order of
15% across 40 cells.
Our simulations have showed that administration of 2-
deoxy-glucose at 3 mM reduces ATP production levels in
tumor cells in outer rim from 7.4 μM/s to 3.8 μM/s and
hypoxic core from 2.7 μM/s to 0.9 μM/s and a loss of
approximately 3,500 cells due to energy depletion (ATP
production levels lower than 8.5 μM/s).
Previous experiments [36] have shown in vitro that the
use of 3 mM 2-deoxy-glucose decreases ATP levels in the
order of 60% in normoxic conditions and 70% in hypoxic
conditions, our simulations showed a decrease in 50%
under normoxic conditions and 70% in hypoxic, showing
agreement with in vitro experimental data.
Control tumor
Without treatment, the tumor reached the lethal size
(10,000 cells in simulated disc corresponding to 700,000
in spherical volume) in 26 generations, equivalent to 6
months. During free growth, fast proliferating chemosen-
sitive cells are selected and thus the average chemoresis-
tance of the tumor decreases gradually while proliferative
phenotype value increases (figures 2 and 3).
Figure 4 shows the morphology of this untreated tumor,
with a few chemoresistant cells in center while most of
tumor mass is composed of chemosensitive cells. The
sensitive population, however, is composed of other sub-
populations with different phenotypes for glucose metab-
olism and acid resistance, some proliferating while others
are quiescent.
MTD
The use of high dose chemotherapy, equivalent to 5 times
IC50 of chemoresistant population and 500-fold higher
than IC50 of chemosensitive cells leads to death of chemo-
sensitive cells and consequent tumor size reduction (fig-
ure 2), but also an increase in the average tumor
chemoresistance due to increased population share of
chemoresistant cells (figure 3).
The repeated use of this strategy eventually creates a
chemoresistant tumor (figure 5) that reaches lethal size in
183 generations (approximately 3.5 years), with low
response to therapy.
2DG only
2-deoxy-glucose administered alone was capable of
reducing the number of cells in tumor short after admin-
istration (5,000 cells to 1,500) without increasing
chemoresistance. However, all dead cells were located in
tumor interior (figure 6), and thus were not in prolifera-
tive state due to lack of free space to grow into. The exter-
nal tumor volume, thus, and its growth rate were only
moderately affected by treatment and the lethal tumor
size was reached in 57 generations (1 year).
Figure 2 Tumor growth for untreated (CONTROL), maximum tol-
erated dose treatment (MTD) and 2-deoxy-glucose only treat-
ment (2DG). Tumor growth for untreated (CONTROL), maximum
tolerated dose treatment (MTD) and 2-deoxy-glucose only treatment
(2DG). Sizes are normalized to 1 being maximum tumor burden. Un-
treated tumors reach limit size after 26 generations; 2DG reaches this
size after 57 generations and MTD after 183 generations.
Figure 3 Average tumor chemoresistance for untreated (CON-
TROL), maximum tolerated dose treatment (MTD) and 2-deoxy-
glucose only treatment (2DG). Average tumor chemoresistance for
untreated (CONTROL), maximum tolerated dose treatment (MTD) and
2-deoxy-glucose only treatment (2DG). Only tumor treated with che-
motherapy will "develop" chemoresistance to treatment as the treat-
ment will change adaptive landscape of tumor to favor cells with
resistant phenotype. In fact, the untreated tumor also contains resis-
tant cells but these are kept in small numbers in absence of therapy.
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Double Bind
For all three scenarios tested of combined therapy, at the
end of 75 generations (~1.5 years) a high intensity dose
was applied to try and eradicate the tumor. This bolus is
identified in figures 7 and 8 as red arrows and was used to
test if the prolonged double bind treatment could eventu-
ally eradicate the chemoresistant subpopulation to the
point that a high dose treatment could eventually lead to
Figure 4 Growth pattern of untreated tumor at generation 39. Growth pattern of untreated tumor at generation 39. A shows chemosensitive cells
(green) and chemoresistant cells (red). B shows distribution of glycolytic phenotype: cells closer to green have slower metabolic rate and those closer to
red are more glucose avid. C shows distribution of acid resistance, the closer to red the more resistant and the ones closer to green are more sensitive to
low pHe. D shows cells in low pHe induced quiescent state in green and proliferative cells in yellow.
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the possibility of cure even in a tumor originally with
chemoresistant cells.
Synchronized
The simultaneous administration of 2-deoxy-glucose and
chemotherapy led to better results than chemotherapy
alone, with most of dead cells being the ones in the outer
rim of tumor, and thus environmentally sensitive. The
lethal size was reached after 210 generations (4 years), an
increase in 15% of life span with reduced chemotoxicity
thanks to lower bolus density (20% of MTD).
The overall size of tumor is kept stable for longer (fig-
ure 7), in contrast with MTD which drastically reduces
tumor size initially but then allows its regrowth. Unfortu-
nately, this treatment also causes rise of chemoresistance
(figure 8) that will cause tumor to cease to respond to
therapy.
The morphology of a tumor treated with synchronized
therapy (figure 9) shows how the chemoresistant popula-
tion is able to break free from tumor center but still has to
compete with a population of sensitive cells.
Chemotherapy followed by 2DG
Chemotherapy followed by glucose competitor leads to
better results than the simultaneous administration, with
smaller tumor size during treatment (figure 7), lower
chemoresistance (figure 8) and a life span of 600 genera-
tions (~11 years) representing and improvement of 3-fold
as compared to MTD.
The better performance of this approach can be
explained by the morphology of figure 10 showing how
sensitive cells are kept at a higher proportion if compared
to resistant cells. This sensitive population is thus more
capable of surrounding resistant cells and slow down
their growth.
Figure 11 shows the comparison of these two combined
treatments and how in the case of synchronized therapy,
resistant cells are left with more space to regrow while in
Figure 5 Growth pattern of a tumor treated with maximum tolerated dose at generation 45. Growth pattern of a tumor treated with maximum
tolerated dose at generation 45. A represents chemoresistance (red resistant and green sensitive), B represents hyperglycolysis (red high glycolysis
and green low), C shows acid resistance (red resistant to low pHe and green sensitive) and D shows quiescent cells (green) and proliferative (yellow).
Despite being 10 times smaller than the original tumor, it is entirely composed of chemoresistant cells (red) and will not respond to therapy.
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the Rx => 2DG the extra generation between two chemo-
therapeutic sessions allows sensitive cells to regrow and
trap resistant ones.
2DG followed by chemotherapy
The use of 2-deoxy-glucose followed by chemotherapy
presented the most promising results. While reducing
chemoresistance (figure 8), tumor size was kept stable
(figure 7) and at the end of simulations, at generation 72,
no more resistant cells were present.
Figure 11 shows a sequence of steps of original tumor
being treated first with 2-deoxy-glucose and next with
chemotherapy. This sequence describes how this
approach is different from previous ones, as the cells in
Figure 6 Growth pattern of 2DG treated tumors at generation 57. Growth pattern of 2DG treated tumors at generation 57. A represents chemore-
sistance (red resistant and green sensitive), B hyperglycolysis (red high glycolysis and gree n low), C shows acid resistance (red high resistance to low pHe
and green sensitivity) and D shows quiescent (green) and proliferative (yellow) cells. Treatment with 2DG may lead to eradication of chemoresistan t cells
but does not prevent tumor growth.
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center of tumor are first led to energy depletion and then
chemotherapy acts on killing the ones in outer regions.
The morphology of this tumor (figure 12) also shows
how this approach pulverizes the tumor in smaller vol-
umes with higher surface-to-volume ratio, increasing
efficiency of chemotherapy. This pulverized morphology
is result of 2-deoxy-glucose energy depletion, preventing
tumor cells from existing in hypoxic environments.
Final Bolus
To asses if tumors can be eradicated after 1.5 years of
treatment, a high dose bolus of MTD was used in three
cases of combined therapy. The three tumors responded
with decrease in volume: 0.35 to 0.03 for synchronized,
0.49 to 0.01 for chemotherapy followed by 2DG (Rx =>
2DG) and 0.19 to 0.002 for 2DG followed by chemother-
apy (2DG => Rx).
The tumor was virtually eradicated for the last strategy
with 22 cells left, all sensitive to therapy, while chemore-
sistance was increased in the two others.
Discussion
This work confirms the prediction of previous mathemat-
ical models and simulations [18,38] that suggested that
administration of chemotherapy with the goal of tumor
stabilization instead of eradication would yield better
results (longer subject survival) than the use of maximum
tolerated doses.
This strategy assumes that resistant cells are already
present in the tumor and will replace the sensitive cells
killed during treatment. This scenario is impervious to
standard MTD treatment and is the strength of this inno-
vative strategy. However, the current models do not take
into account the potential mutagenic effects of the che-
motherapy - a process that could increase the evolution
of resistance. This will be the subject of future work.
The simulations described in this work demonstrate
that chemoresistant cells are eventually trapped inside
tumor as faster replicating sensitive cells outgrow them.
The sensitive cells then will serve as a shield holding these
resistant cells at bay until they are killed by chemotherapy
and the resistant cells are again free to proliferate.
If only the minimum necessary number of sensitive
cells is killed by chemotherapy, a sufficient number will
be available to outgrow the resistant population and keep
it trapped inside tumor.
These simulations confirm that the time for reaching
steady-state distribution of drug in solid tumors is in the
same order as half-life of chemotherapeutics. Figure 13
shows the effect of exposing the same tumor to same
bolus intensity of 2-deoxy-glucose (3 mM) and chemo-
therapy (5 μM) during 1,000 and 5,000 seconds respec-
tively. While shorter exposition was able to kill most of
cells in outer rim of tumor, only the longer exposure was
capable of affecting the chemoresistant cells, and break
up the tumor morphology.
Research with liposomes [30] showed how increasing
half-life of drugs in body can significantly increase its effi-
ciency, especially drugs that target the cells within tumor
core, such as 2-deoxy-glucose.
Our models also address another major barrier for che-
motherapy effectiveness - the reduced sensitivity of qui-
escent cells in interior of avascular or poorly perfused
tumors. Because the drug concentrations vary up to two
orders of magnitude [37] between healthy tissue and
tumor center during transients, even the maximum toler-
Figure 7 Comparison of tumor size in different combined thera-
pies. Co mparis on of tu mor siz e in dif ferent combin ed ther apies . SYNC
stands for simultaneous administration of 2DG and Rx, 2DG => Rx cor-
responds to administration of 2DG followed by Rx, and Rx => 2DG cor-
responds to administration of Rx followed by 2DG. Red arrows
represent final high Rx dose of MTD in one unique dose in order to try
tumor eradication.
Figure 8 Comparison of tumor average chemoresistance in dif-
ferent combined therapies. Comparison of tumor average chemore-
sistance in different combined therapies. Chemoresistance steadily
increases in simultaneous administration, but still more slowly than
MTD. After final high dose bolus only 2DG => Rx strategy's tumor re-
mains sensitive. Legend keys ar e SYNC for simultaneous administration
of 2DG and Rx, 2DG => Rx for administration of 2DG followed by Rx,
and Rx => 2DG corresponds to administration if Rx followed by 2DG.
Red arrows represent final high Rx dose of MTD in one unique dose in
order to try tumor eradication.
Silva and Gatenby Biology Direct 2010, 5:25
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Page 11 of 17
ated dose may be too low to kill the tumor cells within
this region of microenvirnmental resistance.
The use of 2-deoxy-glucose to target cells in ischemic
and hypoxic regions of a tumor was proposed almost
three decades ago but to our knowledge, no quantitative
analysis had been performed on the effect of 2DG on
tumor metabolism and on optimal strategies to exploit
synergy between chemotherapy and energy restriction.
Our simulations indicate that the simultaneous admin-
istration of chemotherapy and 2-deoxy-glucose, the
approach used in prior trials, does not optimize treat-
ment outcome because, as both drugs diffuse through
tumor towards its center, the tumor cells on the outer rim
start dying due to chemotreatment. As these cells die,
their mitochondria shut down and more oxygen and glu-
cose can diffuse towards tumor center, cancelling the
effect of 2-deoxy-glucose.
The simulation's results suggest that 2-deoxy-glucose
has its maximum effect when the tumor exterior is intact
and there is a minimum number of cells between the
Figure 9 Morphology and phenotypic distribution in tumor treated with chemotherapy in tandem with glucose competitor at step 75. Mor-
phology and phenotypic distribution in tumor treated with chemotherapy in tandem with glucose competitor at step 75. A represents chemoresis-
tance (red resistant and green sensitive), B represents hyperglycolysis (red high and green low glycolysis), C shows acid resistance (red for resistance
to low pHe and green for sensitivity) and D shows quiescent cells (green) and proliferative (yellow). The chemosensitive population is not capable of
completely engulfing resistant population what allows faster remission.
Silva and Gatenby Biology Direct 2010, 5:25
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Page 12 of 17
healthy vascularized tissue and the targeted tumor cells
in order to its energy depletion effect to act.
This conclusion inspired us to simulate administration
first of the glucose competitor followed by chemotherapy.
This strategy showed its benefits when at first the tumor
center was replaced by a core of starving dying cells -
including many of the chemoresistant cells- and then had
its exterior rim pulverized by chemotherapy: the repeti-
tion of this treatment eventually led to a tumor with no
chemoresistant population and thus potentially curable.
Conclusions
These results can be generalized into an important con-
clusion that the optimal therapy must look beyond speci-
ficity and immediate cytotoxicity of a chemotherapeutic
drug. The strategy for the treatment must include under-
standing of the evolution and the mechanism for resis-
Figure 10 Morphology and phenotypic distribution in tumor treated with chemotherapy followed by glucose competitor at step 72. Mor-
phology and phenotypic distribution in tumor treated with chemotherapy followed by glucose competitor at step 72. A represents chemoresistance
(red for resistant and green for sensitive), B represents hyperglycolysis (red for high and green for low glycolysis), C shows acid resistance (red for re-
sistance to low pHe and green for sensitivity) and D shows quiescent cells (green) and proliferative (yellow). The extra generation without chemother-
apy allows regrowth of sensitive population enough to encompass and delay growth of resistant cells.
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Page 13 of 17
tance and explicit inclusion of strategies to suppress or
exploit these evolutionary dynamics.
This work proposes that most promising therapies
should be those that manipulate the evolutionary forces
that enforce tumor cell fitness towards making these cells
less fit than normal cells, as described in this work.
The robustness of tumors comes in part from its
genetic and phenotypic variation but also from its envi-
ronmental heterogeneity. So far, chemotherapy and
radiotherapy specialists have faced this heterogeneity as
obstacles to be overcome while we propose to exploit
these properties and use environmental barriers to trap
the tumor in its own microenvironment and reduce its
resistance to treatment.
Appendix A: Computational Model
The solid tumor model for growth and chemotherapy
response was simulated using TSim software http://
www.i-genics.com, which is available upon request. This
software was developed ad hoc for simulating biological
systems with cells interacting in a microenvironment.
This model was represented as a fixed-lattice 3D cellu-
lar automaton where each cell has its own set of inherit-
able phenotypes. All cells are incompressible and
represented as cubic volumes with 25 μm of side. Cell
motility was also abstracted and the "movement" of the
tumor is based solely on cell replication and death.
The simulation is composed of three steps: the first
consists in cells dying or replicating at every generation
(weeks between each of them); each replication or cell
death decision is taken based on extracellular concentra-
tion of species and intracellular ATP production rate. The
extracellular concentration of species depends on the
equilibrium between the production/consumption of
these species and the diffusion of these species from
blood vessels where they are produced (O2, glucose,
bicarbonate anions) or consumed (H+, CO2).
Each simulation consisted in multiple generations, each
generation corresponding to one week, enough for the
model to reach steady-state between metabolism and dif-
fusion of species between two generations. This steady
state was calculated in a transient of 5,000 steps, each one
corresponding to 1 second in order to simulate the tran-
sient of drug concentration in blood and the diffusion of
drug into tissue and tumor.
The diffusion of species was calculated numerically
using the heuristic described in [23].
Vascularization was represented as blood vessels form-
ing a cubic mesh at a distance of ~4 mm (150 cells) cells
from the tumor's center. This vascularization was used as
boundary condition: in these vessels oxygen partial pres-
sure, glucose concentration, pH and bicarbonate concen-
trations were kept at a fixed value.
Cells can proliferate, die or remain quiescent depend-
ing on their ATP production rate or extracellular pH
(pHe). Should the pHe value be below 7.1 (for acid sensi-
tive cells) or 6.9 (for acid resistant cells), the tumor cells
would become quiescent and would not replicate until
the next cell cycle when energy and pHe restrictions are
satisfied. Tumor cells would die when exposed to pHe
below 6.8 (for acid sensitive) or 6.6 (for acid resistant).
At every generation the probability of a cell replication
increases as its ATP production is above a minimum level
of 0.85 uM/s [28] and plateaus at 8.5 uM/s, for ATP pro-
duction rates lower than the minimum value the cell dies
due to starvation as described in equation 1:
Each cell possess its own energetic metabolism simpli-
fied to aerobic conversion of one molecule of glucose into
36 molecules of ATP and 6 of CO2 or anaerobically into
pATP ATP
ATP
p
DUPLICATION
D
=−×
××≤
≤
(. )
..
01 0
09 0
01
0
If ATP ATP
UUPLICATION
DUPLICATION
p
=≤×
=≤
001
1
0
0
If ATP ATP
If ATP ATP;
.;
Figure 11 Three different ways of combining chemotherapy and
2-deoxy-glucose. Three different ways of combining chemotherapy
and 2-deoxy-glucose: (a) First chemotherapy and later 2DG, (b) 2DG
first followed by chemotherapy and (c) both administered at the sa me
time. The first approach kills most of the sensitive cells of tumor but
misses some of the resistant ones, as a consequence, the tumor perfu-
sion is improved and more oxygen and glucose is available to the re-
maining cells, neutralizing the effect of 2DG. The simultaneous
administration of Rx and 2DG leads to the same scenario, with sensitive
and resistant cells left in a more oxygenated porous tumor mass. In
both scenarios, with resistant cells free from the tumor core where
they were trapped, the fraction of resistant cells will grow and tumor
will slowly become more resistant. The strategy that is rewarded with
most benefits is the administration of glucose competitor followed by
chemotherapy, what produces a smaller sensitive tumor.
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Figure 12 Morphology and phenotypic distribution of tumor treated with 2DG followed by chemotherapy at step 72. Morphology and phe-
notypic distribution of tumor treated with 2DG followed by chemotherapy at step 72. A represents chemoresistance (red resistant and green sensi-
tive), B hyperglycolysis (red high glycolysis and green low), C shows acid resistance (red resistance to low pHe and green sensitivity) and D shows
quiescent cells (green) and proliferative (yellow). The "pulverized" morphology grants a higher surface-to-volume ratio what increases efficiency of
chemotherapy. All resistant cells have been removed during treatment by energy depletion.
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two molecules of ATP and 2 of lactic acid. The anaerobic
or aerobic paths are chosen depending on the availability
of oxygen in extracellular environment according to
equations 2 to 7:
Glucose aerobic metabolism in cytoplasm and mito-
chondria:
Glucose anaerobic metabolism in cytoplasm:
In hypoxic conditions (excess of glucose):
In hyperoxic or hypoglycemic conditions (excess of O2):
ATP ANAER and ATPAER correspond to the production
rate of ATP anaerobically and aerobically respectively and
their sum is the total ATP production rate.
Reviewer's Comments
Reviewer #1: Mark P Little
Although much data is referenced by the authors for the
various model parameters, there is little biological justifi-
Glc O ATP CO+× ⇒ × +×6366
22
(2)
Glc ATP LacticAcid⇒× +×22 (3)
ATP Glc O
ANAER =× −
⎛
⎝
⎜⎞
⎠
⎟
22
6
[]
[] (4)
ATP O
AER =×
⎛
⎝
⎜⎞
⎠
⎟
36 2
6
[] (5)
ATP Glc
AER =×36 [ ] (6)
ATPANAER =0(7)
Figure 13 Drug concentration is not the only factor defining efficiency in tumor cells eradication. Drug concentration is not the only factor
defining efficiency in tumor cells eradication. The diffusion of species from blood vessels to the interior of avascular tumors creates gradients of con-
centration that favors survival of cells further from tumor-host interface. The exposure of the tumor model to 2DG during 5,000 seconds is enough to
produce energy depletion in most cells inside tumor but can only kill a fraction of cells in intermediate region if exposed to only 1,000s. This exposure
time is crucial for the efficiency of a drug such as 2DG which acts mainly in inner regions of tumor.
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cation presented for certain crucial modelling assump-
tions. In particular, fundamental to the model is the
assumption that there are two populations of cells, one
chemo-sensitive and highly proliferative, the other chemo-
resistant and more slowly proliferative. However, little evi-
dence is presented for this crucial assumption. Can the
authors provide such evidence?
We thank Dr Little for his remarks and agree that not
all forms of chemotherapy will incur in a cost of resis-
tance and thus make resistant cells less proliferative. The
hypothetical therapy proposed in this work, however, tar-
gets cancer cells that are more proliferative than normal
host cells. This definition of target automatically confers
resistance to tumor cells that proliferate more slowly.
These cells, even in absence of treatment will be minority
in the tumor (due to their slow proliferation) and will
eventually remain trapped in the interior of the tumoral
mass.
This is a first simplification from this model since there
are other mechanisms of multi-drug resistance- like
efflux pumps, for instance- which may be energetically
taxing but not directly affect the proliferation rate in an
environment rich in nutrients. The second simplification
is that there might be gradients of proliferative and resis-
tant phenotypes, the use of two subpopulations however
allowed us to infer the proportion of resistant and sensi-
tive cells directly from the overall tumor resistance (Fig-
ures 3 and 8).
Reviewer #2: Marek Kimmel
The paper advocates, based on biological observations
and mathematical modeling, a new strategy of tumor che-
motherapy by combination of agents, which jointly protect
the balance between the sensitive and resistant pheno-
types among cells constituting a tumor.
This concept seems to amount to a type of sophisticated
maintenance therapy, the examples of which are not
entirely unknown in current oncology. Perhaps the authors
might discuss some examples of maintenance therapy in
existence, which work in a similar way.
Counterexamples will also be productive for elucidation
of the unique features of the approach proposed.
We thank Dr Kimmel for his remarks and suggestions
on this work.
There are two basic ideas proposed in this article: (1)
the sensitive share of the population of a solid tumor (the
vast majority of the tumor mass in cancers that normally
respond to therapy at a first moment) could be used to
slow down the growth of the resistant mutants and thus
delay recurrence, and (2) the use of currently available
drugs could be optimized if administered in a way that
each drug modifies the tumor microenvironment (tumor
+ stroma + vessels + growth factors) in order to sensitize
it to the next drug.
The first concept was explicitly proposed by Gatenby et
al. (2009) [39] and labelled "Adaptive Therapy", which
advocates the use of the minimum amount of drug
required to keep the total tumor size stable. Computer
simulations and animal models were used to test this con-
cept, which was extended in this work where the simula-
tions include the spatial distribution of these cells missing
in our previous work.
The second concept is slightly different from what is
currently done in clinical trials where different drug com-
binations are tested in order to find protocols that could
delay recurrence in incurable diseases. In Multiple
Myeloma, for instance, there is a school of thought which
proposes to use the maximum number (and amount) of
drugs available in the beginning of the treatment, while a
second opposite approach dictates to save drugs for later
and thus avoid tumors that recur resistant to all currently
available drugs [40]. What is missing in this discussion,
however, are studies showing exactly how the disease
progresses after treatment with each drug, how the sub-
populations of the primary tumor, metastases and circu-
lating tumor cells are selected and how each of these
microenvironments are changed.
Such studies, however, require realistic quantitative
models of both disease progression and in vivo response
to therapy. Building such models is a great challenge since
most quantitative data (dose response curves) are
obtained in vitro using cell lines, which cannot fully rep-
resent the phenotypic and environmental heterogeneity
of real tumors.
What we show in this work is only a proof of concept of
how much more powerful therapies could become, if only
we knew exactly how the treatment is affecting the
microenvironment and in which direction its is being
driven (towards cure, recurrence or maintenance).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
RAG participated in the elaboration of the biological question and in the
review of the manuscript. AS developed the computer model and wrote the
manuscript.
Acknowledgements
We thank Professor José Andrés Yunes from Centro Infantil Boldrini, for his
remarks and suggestions on the early version of this article.
As a one-time exception to the publishing policy of Biology Direct the articles
in this series are being published with two reviewers.
Author Details
Department of Radiology and Integrative Mathematical Oncology, Moffitt
Cancer Center, Tampa, FL, USA
Received: 31 December 2009 Accepted: 20 April 2010
Published: 20 April 2010
This article is available from: http://www.biology-direct.com/content/5/1/25© 2010 Silva and Gatenby; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Biology Direct 2010, 5:25
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Page 17 of 17
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Cite this article as: Silva and Gatenby, A theoretical quantitative model for
evolution of cancer chemotherapy resistance Biology Direct 2010, 5:25
