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Prediction of drug response in breast cancer using integrative
experimental/computational modeling
Hermann B. Frieboes1, Mary E. Edgerton4, John P. Fruehauf9, Felicity R. A. J. Rose10, Lisa
K. Worrall10, Robert A. Gatenby11, Mauro Ferrari2,3,5,7, and Vittorio Cristini1,3,6,8,*
1School of Health Information Sciences, University of Texas Health Science Center, Houston, Texas
2Division of Nanomedicine, University of Texas Health Science Center, Houston, Texas
3Department of Biomedical Engineering, University of Texas Health Science Center, Houston,
Texas
4Department of Anatomic Pathology, The University of Texas M.D. Anderson Cancer Center,
Houston, Texas
5Department of Experimental Therapeutics, The University of Texas M.D. Anderson Cancer Center,
Houston, Texas
6Department of Systems Biology, The University of Texas M.D. Anderson Cancer Center, Houston,
Texas
7Department of Bioengineering, Rice University, Houston, Texas
8Department of Biomedical Engineering, The University of Texas, Austin, Texas
9Division of Hematology/Oncology, Department of Medicine, University of California, Irvine, Medical
Center, Orange, California
10School of Pharmacy, Centre for Biomolecular Sciences, University Park, University of Nottingham,
United Kingdom
11Departments of Radiology and Integrated Mathematical Oncology, Moffitt Cancer Center, Tampa,
Florida
Abstract
Nearly 30% of women with early stage breast cancer develop recurrent disease attributed to resistance
to systemic therapy. Prevailing models of chemotherapy failure describe three resistant phenotypes:
cells with alterations in transmembrane drug transport, increased detoxification and repair pathways,
and alterations leading to failure of apoptosis. Proliferative activity correlates with tumor sensitivity.
Cell cycle status, controlling proliferation, depends upon local concentration of oxygen and nutrients.
Although physiological resistance due to diffusion gradients of these substances and drug is a
recognized phenomenon, it has been difficult to quantify its role with any accuracy that can be
exploited clinically. We implement a mathematical model of tumor drug response that hypothesizes
specific functional relationships linking tumor growth and regression to the underlying phenotype.
The model incorporates the effects of local drug, oxygen and nutrient concentrations within the three-
dimensional tumor volume, and includes the experimentally observed individual cells’ resistant
phenotypes. By extracting mathematical model parameter values for drug and nutrient delivery from
monolayer (one-dimensional) experiments and using the functional relationships to compute drug
*Correspondence: Vittorio Cristini, University of Texas HSC—SHIS, 7000 Fannin #600, Houston, TX 77030, USA. Tel.: 713-500-3965;
Fax: 713-500-3929; E-mail: E-mail: vittorio.cristini@uth.tmc.edu.
NIH Public Access
Author Manuscript
Cancer Res. Author manuscript; available in PMC 2010 May 15.
Published in final edited form as:
Cancer Res. 2009 May 15; 69(10): 4484–4492. doi:10.1158/0008-5472.CAN-08-3740.
NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript
delivery in MCF-7 spheroid (three-dimensional) experiments, we use the model to quantify the
diffusion barrier effect, which alone can result in poor response to chemotherapy both from
diminished drug delivery and from lack of nutrients required to maintain proliferative conditions.
We conclude that this integrative methodology tightly coupling computational modeling with
biological data enhances the value of knowledge gained from current pharmacokinetic
measurements, and, further, that such an approach could predict resistance based on specific tumor
properties and thus improve treatment outcome.
Keywords
breast cancer; drug response; mathematical model
INTRODUCTION
We implement a novel quantitative approach that links growth and regression of a tumor mass
to the underlying phenotype to study the impact of drug and nutrient delivery as mediators of
physiological resistance. It is well known that inefficient vascularization may prevent optimal
transport of oxygen, nutrients, and therapeutics to cancer cells in solid tumors [1]. As a result,
the drug agent must diffuse through the tumor volume to reach the entire tumor cell population,
and there is mounting evidence to indicate that these diffusion gradients may significantly limit
drug access [2–5]. Both hypoxia and hypoglycemia contribute to physiological resistance
through various mechanisms, including induction of oxidative stress and a decrease in the
number of proliferating cells [6–12]. The myriad of stresses can lead to selection of cells that
resist apoptotic conditions, thus adding to pathologic resistance in tumors [13]. This evidence
strongly suggests that the diffusion process alone can lead to the evolution of drug resistance
in tumor cells that exceeds predictions based on individual cell phenotype [5]. It has not been
easy to quantify the resistance effects of diffusion gradients with any accuracy that can be
exploited in a clinical setting. The different physical scales in a tumor spanning from the nano-
to the centimeter scale present a complex system that to be better understood could benefit
from appropriate mathematical models and computer simulations in addition to laboratory and
clinical observations.
In particular, biocomputational modeling of tumor drug response has endeavored in the last
two decades to address this need. Space limitations preclude a full description (see [5,14,15],
references therein). Dox cellular pharmacodynamics has been modeled; e.g., [16] presented a
model providing good fits to in-vitro cytotoxicity data. Drug transport was modeled in
spheroids vs. monolayers [17]. A model capable of predicting intracellular Dox accumulation
that matched experimental observations was described in [18]. Different drug kinetics effects
in-vitro were compared in [19], showing that a single drug infusion could be more effective
than repeated short applications. Models employing multi-scale approaches, i.e., linking events
at sub-cellular, cellular, and tumor scales (e.g., [20]), studying vascularized tumor treatment
(e.g., [21]), and simulating nanoparticle effects (e.g., [3]) have also been developed. Existing
mathematical models are often limited to radially-symmetric tumor representations and not
fully constrained through experimentally-set parameters.
Here, we employ a multi-scale computational model, extending a previous formulation of
tumor growth founded in cancer biology [22–25], to enable more rigorous quantification of
diffusion effects on tumor drug response. This model can represent non-symmetric solid tumor
morphologies in 3-D, thus providing the capability to capture the physical complexity and
heterogeneity of the cancer microenvironment. More significantly, we fully constrain the
model through functional relationships with parameters set from experiments. We hypothesize
the simplest relationships that would at the same time be biologically founded and which could
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be calibrated by the experimental data. These relationships link tumor mass growth and
regression to the underlying phenotype. They provide the mathematical basis for describing
cell mitosis, apoptosis and necrosis modulated by diffusion gradients of oxygen, nutrients, and
cytotoxic drugs, and enable quantification of the physiological resistance introduced by these
gradients. Input parameters include the diffusion coefficients for these substances, and the rate
constants for proliferation, apoptosis, and necrosis. We measure parameter values from
independent experiments performed under conditions with no gradients, i.e., with cells grown
as 1-D monolayers, and then use these values to calculate cell survival in 3-D tumor geometry.
These values are compared with experiments in which cells are grown as in-vitro tumor
spheroids, representing a 3-D tumor environment with diffusion gradients. This approach
allows us 1) to fully constrain the computational model, using experimentally-obtained
parameter values, and 2) to validate the hypothesized functional relationships by comparing
the computed 3-D tumor viability with the spheroid tumor growth experiments. By quantifying
the link between tumor growth and regression and the underlying phenotype, the work
presented here provides a quantitative tool to study tumor drug response and treatment.
Although in-vitro spheroids are a gross simplification of the complex in-vivo condition, they
allow capturing the effect of 3-D tissue architecture in generating diffusion gradients of drug
and cell substrates under controlled conditions without the complicating effects of blood flow
[1]. Spheroids develop a layer of viable cells surrounding a necrotic core [26]. The thickness
of this layer (ca. 100 µm), maintained by substrate diffusion gradients, “mimics” the viable
tissue thickness supported by diffusion from surrounding blood vessels in-vivo [26]. This
provides a more controllable experimental model than in-vivo, which is necessary in order to
test and calibrate the computational model parameters. In turn, computer simulations of the
model enable formulating and testing the functional relationships that quantify the dependence
of drug resistance on diffusion gradients and particular tissue characteristics (e.g., tissue
compactness as a function of cell packing density [27]). We conclude that computational
modeling tightly integrated with tumor biology extends the information that can be learned
from pharmacokinetic experiments [28–30], offering a promising possibility of ultimately
quantifying and predicting treatment response from individual tumor characteristics.
MATERIALS AND METHODS
Doxorubicin was used as the cytotoxic drug in the experiments. Experiments used MCF-7 WT
breast cancer cell lines to represent drug-sensitive tumor cells and MCF-7 40F breast cancer
cell lines to represent drug-resistant tumor cells. Details of experimental methods are given
below.
Observation of cell substrate gradients
Confluent MCF-7 WT (breast cancer) cell monolayers were incubated with 0.25% (w/v) trypsin
(Sigma, UK) and 0.02% (w/v) EDTA (Sigma, UK) for 4 minutes at 37°C. Complete media was
added to this mixture to inhibit enzyme activity and the cell suspension passed through a 24-
gauge needle 6 times to ensure a single cell population. A 10 mL High Aspect Ratio Vessel
(HARV; Synthecon, USA) was seeded at 2×106 cells/mL and rotated at 15 rpm at 37°C in a
humidified atmosphere of 5% CO2 in air. Aggregates formed within 6 hours and were
maintained in RCCS culture for a total of 30 days. During this time, media in the HARVs was
replenished 50:50 every other day. Spheroids were fixed in formalin, embedded in paraffin
and 5 µm sections cut for immunohistochemistry. To observe hypoxia, unfixed spheroids were
incubated in 200 µM pimonidazole (Chemicon, UK) in complete media for 2 hours at 37°C
prior to fixation as above. Proteinase K digestion was used for antigen retrieval prior to 1° Ab
incubation. The mouse IgG 1° Ab was used at 1:600 dilution and incubated with 4 µm sections
for immunolocalization of hypoxia through detection of pimonidazole protein adducts.
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Immunohistochemical (IHC) staining for GLUT-1 was performed using rabbit polyclonal
antibody against the C-terminal portion (Abcam). The NHE-1 antibody is a rabbit polyclonal
antibody (Santa Cruz). Immunohistochemistry was performed using the Discovery XT
Automated Staining platform (Ventana Medical Systems, Inc, Tucson, Arizona).
Deparaffinization and antigen retrieval of tissue sections was performed online. Antigen
retrieval was performed on the Ventana instrument with a borate buffer (pH 8) 40 minutes at
95°C. Once the tissue was conditioned in this way, primary antibody was applied manually at
a 1:800 dilution. (60 min. incubation at 37°C). Primary antibodies were visualized using VMSI
validated detection and counterstaining reagents. Images were captured using an Olympus
BX50 camera with an RT SPOT (Diagnostic Instruments, Inc), and standardized for light
intensity.
Drug response
MCF-7 WT (drug-sensitive) and MCF-7 40F (drug-resistant) cell lines were cultured in RPMI
Media 1640 (Life Technologies Invitrogen, Carlsbad, California) supplemented with 3% FBS
(Life Technologies Invitrogen, Carlsbad, California), 2 mM L-glutamine, and 1% penicillin/
streptomycin in humidified 7.5% CO2 at 37°C. Cells for monolayer culture were seeded 20,000
per well in 24-well Costar 3527 cell culture plates (Corning, New York). Cells for spheroids
were seeded 50,000 per well in 24-well Costar 3473 cytophobic plates, and shaken at 100 rpm
for 10 min. on day 1. Both the cells in monolayer and spheroids were incubated for 3 days, and
then exposed to doxorubicin (Dox) (Bristol-Myers Squibb, Princeton, New Jersey)
concentrations ranging from 0 to 16384 nM in 4× nM increments (0, 4, 16, 64, etc.) for 96
hours, representing at 256 nM a typical area-under-the-curve1 in-vivo [29]. Negative controls
were seeded and incubated under the same conditions without drug. Three endpoints were
concurrently measured as fraction of negative control: proliferation using tritiated thymidine
(Amersham, Buckinghamshire, Great Britain) incorporation assay [31], viability using trypan
blue exclusion counts, and metabolic activity using XTT (sodium 3'-[1-[(phenylamino)-
carbonyl]-3,4-tetrazolium]-bis(4-methoxy-6-nitro)benzene-sulfonic acid hydrate) assay [32].
All experiments were done in triplicate. Photographs were taken with a digital camera through
a Zeiss microscope (100× magnification).
Mathematical model of tumor growth and drug response
Briefly, the equations [22,23] are formulations of mathematical models used in engineering to
describe phase separation of two partially miscible components (viable and necrotic tumor
tissue), diffusion of small molecules (cell substrates and drug), and conservation of mass
(Quick-Guide, Supplementary Figure 1, see also [33–36]). Mass conservation equations
describe growth (proliferation as a function of total number of cycling cells) and death from
the drug’s cytotoxic effects (apoptosis as a function of a rate constant dependent upon the
unique cell sensitivity and as a function of cycling cells). These are combined with diffusion
of small molecules to a reaction-diffusion equation. Rate constants for proliferation and
apoptosis are modified by functions that represent their dependence upon cell nutrients and
oxygen (proliferation) and drug concentration (death), along with a dependence on spatial
diffusion of these substances. By dimensional analysis of the reaction-diffusion equation the
diffusion penetration length L = (D/η)1/2, where D is the corresponding diffusion constant and
η is a characteristic cellular uptake rate. Large differences in concentration will occur if L is
much less than the tumor radius (linear dimension) in the absence of blood vessels, creating a
steep gradient. By using L and knowing the spheroid size, one can determine the steepness of
these gradients by solving the reaction-diffusion equation. We note that gradients are affected
1In pharmacokinetics, the area-under-the-curve (AUC) is a calculation to evaluate the body's total exposure to a given drug. In a graph
plotting plasma drug concentration vs. time, the area under this curve is the AUC
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by cell packing density, intra-cellular uptake, and duration of drug exposure, as well as
conditions specific to particular experiments.
Estimation of mathematical model parameters
In our model, cell substrates are represented by glucose and oxygen with L estimated from
independent measurements taken from the literature. For glucose D ~ 1×10−7 cm2/sec [37] and
uptake rate may be as large as 1×10−3 sec−1 [38], while for oxygen the corresponding values
are ~ 1×10−5 cm2/sec [39] and 1×10−1 sec−1 [40]. Diffusion penetration lengths thus calculated
(L ≈ 100 µm) are consistent with our experimental observations (IHC) and with previous
(murine mammary tumor EMT6/Ro spheroid) observations showing glucose concentration
decreasing by 65% [41] and oxygen decreasing by 90% [42] across the tumor viable region.
Although gradients of drug into tumor tissue have been observed with a number of different
cell types, many of these observations have been qualitative. Thus, published values for
diffusion constants and uptake rates can vary considerably. We estimate Dox diffusion length
by considering that the distance at which the penetrating concentration will be 50% of the
source is ln 2 times the diffusion length – equivalent L here is ca. 90 microns for a 2/3 drop
across the spheroid viable region (~ 100 µm). This is consistent with in-vitro Dox penetration
into Chinese hamster lung cell spheroids at 24 hours [43] reporting an average 2/3 drop of
external concentration across the viable region. Other studies have shown Dox penetration in-
vivo (mouse model) decreasing by half within 40–50 µm of blood vessels by 3 hours [4], as
well as penetration in humans (biopsies) decreasing by half within 50 µm of blood vessels after
2 hours of a bolus and within 60–80 µm after 24 hours in breast cancer tissue [2]. By fitting
the solution of the unsteady Eq. (3) to this data we could indirectly estimate average cellular
uptake rates η of Dox by breast cancer cells in 3-D tissue of ~ 1 day−1 , consistent with the
independent penetration length estimate L = (D/η)1/2 from the steady-state profiles.
The rate λM (inverse time) measures the change in cell number in a population due to mitosis,
normalized by total cell number (control = cells with no drug), and thus was calibrated by
matching proliferation data from our experiments, using as initial guess for λM/λM,C, the in-
vitro cell proliferation as fraction of control divided by in-vitro cell viability (total number of
cells N in monolayer) as fraction of control, yielding an estimate of cell proliferation ~1
day−1.
Apoptosis rate λA (inverse time) over the period 0 < t < T (total experiment time) was estimated
as a function of local concentrations of substrate σ and drug d as
and . Death rate λA’ was calculated from measurements of cell
viability N as fraction of negative control NC over time in monolayer cell cultures at various
drug concentrations. Physiological resistance based on cell cycle status was modelled with
λA". Decreased substrate concentration in 3-D results in cell populations that cycle less and
therefore have reduced sensitivity to Dox2. In the above formula, enhanced survival was
assumed linearly dependent on substrate concentration with a fitting constant calibrated from
previous observations [12] showing an average MCF-7 viability increase 2.2 to 4 times for
glucose deprivation of up to 50% at drug concentrations similar to our study, i.e., 2.2≤1+α/
2Cytotoxicity of MCF-7 cells to Dox may be affected more by glucose deprivation than by hypoxia [6,44]. The glucose-regulated stress
response [10] has been correlated with resistance to topoisomerase II-directed chemotherapeutic drugs such as Dox [10,11] through a
decreased expression of topoisomerase II in MCF-7 cells [12].
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2≤4.0. In our experiments, glucose concentration in the medium was σ∞ = 2.0 g/L. Note that
for σ = σ∞, the above formula gives and thus .
Necrosis parameters for diffusion-limited growth λN (rate of necrosis) and σN (substrate limit
for cell viability) were calibrated by matching simulation results to in-vitro spheroid growth
curves and histological data under conditions with no drug, as described previously [45], i.e.,
so that the simulated spheroid radius and viable region thickness matched in-vitro observations.
These parameters are directly responsible for the steady spheroid size (and extent of necrosis)
after a growth period because they regulate the balance of mass growth due to cell proliferation
in the viable region and mass loss due to cell disintegration in the necrotic center [45]. Based
on our in-vitro measurements of spheroid and necrotic volumes we found a stationary average
spheroid radius of ca. 0.8 mm, with viable region of thickness ~0.1 mm (see also [46]). The
latter is consistent with the estimated cell substrate penetration length L calculated above. The
model calibration based on these quantities consistently yielded:λN/λM = 0.7 and σN/σ∞ = 0.5.
Higher cell densities were observed for resistant spheroids, which had a more compact
morphology [27]. Based on these observations, higher cell adhesion parameter values [22]
were used in the model equations to simulate resistant spheroids than for sensitive ones. We
used a recently presented calibration procedure [45] to determine the relative values for
sensitive and resistant cell cultures. This observation and the resulting different parameters for
the two cell phenotypes underscore the need to explicitly model cell adhesion forces when
constructing models of cancer growth.
QUICK-GUIDE TO EQUATIONS AND ASSUMPTIONS
Equation (1)
The 3D in-vitro environment is modeled as a mixture of viable tumor tissue (volume fraction
ϕV), dead tumor tissue (ϕD), and interstitial fluid (ϕW, given indirectly by 1−(ϕV+ϕD) and
assumed to move freely) flowing through the extracellular matrix, which we treat as a porous
medium. These partial differential equations are derived from the conservation of mass of these
quantities [22,23]. From left to right in each equation, the terms represent: change of volume
fraction with respect to time; tissue advection (bulk transport) by the overall mixture as it moves
with local velocity u; net tissue diffusion due to the balance of mechanical forces, including
cell-cell adhesion (related to mobility constant M [22,23]), and cell-cell adhesion and repulsion
(oncotic pressure), modeled using the variational derivative of the adhesion energy E (for
specific forms of this energy, see [22]); and net source of tissue due to the balance of mitosis,
apoptosis, and necrosis.
In Words
The temporal rate of change in viable and dead tumor tissue at any location within the tumor
equals the amount of mass that is pushed, transported, and pulled due to cell motion, adhesion,
and tissue pressure, plus the net result of production and destruction of mass due to cell
proliferation and death.
Major Assumptions of the Model
The tumor is a mixture of cells, interstitial fluid, and extracellular matrix (ECM), and we treat
the motion of interstitial fluid and cells through the ECM as fluid flow in a porous medium.
Therefore, no distinction between interstitial fluid hydrostatic pressure and mechanical
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pressure due to cell-cell interactions is made. Cell velocity is a function of cell mobility and
tissue oncotic (solid) pressure (Darcy’s law); cell-cell adhesion is modeled using an energy
approach from continuum thermodynamics [22].
Equation (2)
These equations specify net sources SV and SD of mass for viable and dead tumor tissue,
respectively. This directly links the conservation of mass equations Eq. (1) to the cell phenotype
through hypothesized phenomenological functional relationships that include diffusion of cell
substrates and drug (of local concentration σ̣ and d, respectively) through tumor interstitium
(see [22] and references therein). For the viable tissue (first equation), the terms on the right
hand side represent, from left to right, volume fraction gained from cell mitosis (rate λM), and
lost to cell apoptosis (λA) and necrosis (λN). For the dead tissue (second equation), the terms
on the right hand side represent, from left to right, increase in volume fraction from cell
apoptosis and necrosis, and decrease in volume fraction from cell disintegration by lysis (λL).
All rates are inverse time.
In Words
Mass of viable tumor tissue increases through cell proliferation and decreases through cell
apoptosis and necrosis. Mass of dead tumor tissue increases through cell apoptosis and necrosis,
and decreases through cell lysing. These equations provide a means of incorporating the
biology of the problem into the physical conservation laws in Eq. (1).
Major Assumptions of the Model
Cells are composed entirely of water, which is a reasonable first approximation in terms of
volume fraction (see discussion in [22]). Cell mitosis is proportional to substrates present
[47]. As mitosis occurs, an appropriate amount of water from the interstitial fluid is converted
into cell mass. Substrate depletion below a level σN leads to necrosis [26,7]. Cell lysis
represents a loss of mass as cellular membranes are degraded and the mass is converted into
water that is absorbed into the interstitial fluid. The simulated interface between viable and
necrotic tissue is forced to be of biologically realistic thickness (10–100 µm) through the
Heaviside function ℋ (see [22,23]. Mitosis and apoptosis rates are functions of drug (d) and
substrate (σ) concentrations. Note that the assumption that parameters measured for drug
sensitivity in monolayer, modulated by diffusion gradients, can adequately represent response
in-vivo may not be correct for real tumors; e.g., it disregards drug resistance from growing with
an ECM.
Equation (3)
This is a partial differential equation that describes cell substrate concentration σ across the
tumor viable region. The first term on the right hand side models diffusion of the substrates
into the tumor spheroid with penetration length L. The second term models substrate uptake
by the cells (with rate η). An analogous equation is posed for drug concentration d.
In Words
The temporal rate of change of cell substrates or drug across the tumor viable region equals
the net amount that diffuses into the region minus the amount uptaken by the tumor cells.
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Major assumptions of the Model
Diffusion of cell substrates and drug into the tumor, combined with uptake in the tumor interior,
creates and maintains a gradient of these substances through the 3-D tumor tissue, which is
assumed to be fairly compact (non-invasive/infiltrative).
RESULTS
The hypothesized functional relationships of the computational model link growth and
regression of the tumor mass to the underlying phenotype as described in Eq.1–Eq.3 (Quick-
Guide). The model is fully constrained (Methods) through experimentally-based parameters
(Figure 1). In order to quantify physiological resistance in 3-D tumor tissue, diffusion gradients
are incorporated in the functional relationships and the model is calibrated to compute these
gradients. Accordingly, immunohistochemical measurements were performed on drug-
sensitive tumor spheroids cultured in-vitro (Figure 2) to confirm that in our experiments 1) cell
substrate concentration decreases across the radius of the spheroid, 2) apoptosis correlates with
hypoxia, and 3) proliferation correlates with substrate availability [43,47]. Increasing positivity
for pimonidazole protein adducts across the viable rim (~ 100 µm) in the direction of the
spheroid center indicates a gradient of oxygen. Across the viable region, upregulation of Na+/
H+ transporters detected by increasing positivity for NHE-1 is observed, demonstrating a
gradient of pH. Na+/H+ transporters are upregulated in response to acidosis. Upregulation of
GLUT-1 transporters (adaptation to hypoxic/hypoglycaemic stress) is shown in the increasing
positivity for GLUT-1 [48] in peri-necrotic regions, indicating gradients of glucose and oxygen.
The frequency of apoptotic cells increases towards the center as shown by increasing positivity
of TUNEL3 staining. End stage apoptotic bodies are concentrated at the peri-necrotic necrotic
regions with most cells in the center being apoptotic or necrotic. Cell proliferation is limited
to the viable rim, as demonstrated by Ki-67 nuclear staining.
According to the computational model, the solution of the reaction-diffusion Eq. (3) yields
substrate gradients (Figure 3A) developing across a viable region of width ~ 0.1 mm. The
computed width of this region is consistent with previous observations by other investigators
(e.g., [26]). Both glucose and oxygen substrate concentrations were calculated in the model to
drop by at least 50% across the spheroid viable region, with an average concentration 〈σ/σ∞〉
≈ 0.7 in the living tumor tissue. This is in agreement with measurements of glucose [41] and
oxygen (e.g., [42]) in spheroid viable regions from previously reported in-vitro measurements
(Methods).
Similarly, the model computes the drug gradients that develop within the viable tumor spheroid
tissue. The calculated drug concentration profile reaches steady state in ca. 24 hours and decays
by 2/3 across the viable region to yield an average drug concentration of 50% that of outside
the spheroid (Figure 3B). Note that the drug penetration length is comparable to the penetration
length of substrates (Figure 2). Since Doxorubicin is a cell–cycle specific drug and the
proportion of cells that are cycling correlates with the substrate availability, the drug would be
less effective as the proportion of cycling cells is diminished. Not only is its concentration
decreased by the diffusion barrier, but its efficacy is impaired as a result of the substrate
gradients. The physiologic resistance predicted by the model based on the hypothesized
functional relationships would in principle apply to any cell-cycle specific drug.
Cell proliferation (ratio of proliferation-to-viability counts) in-vitro under conditions with
diffusion gradients (i.e., in spheroids) vs. monolayers was lower by at least 50% for both cell
lines at drug concentrations d > 64 nM representing clinically relevant dosages4 (Figure 4A).
3Terminal deoxynucleotidyl Transferase Biotin-dUTP Nick End Labeling.
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Further, increased viability for drug-sensitive cells in 3-D corresponded with ca. 50% lower
cell metabolic activity (ratio of metabolic-to-viability counts) in-vitro (data not shown). These
data indicate that cell quiescence was significant at drug dosages similar to in-vivo conditions.
The cell apoptosis parameter of the computational model calculated from the monolayer cell
viability data increases with drug concentration (Figure 4B).
The functional relationships of the computational model (Eq. 1–Eq.3) linking growth and
regression of the tumor mass to the underlying phenotype are validated by quantifying the
physiological resistance introduced by the diffusion gradients. Cell viabilities predicted by the
model agree well with the ones observed experimentally in 3-D spheroids (Figure 5). In
contrast, monolayer cell viability data is a poor prognosticator of drug resistance in 3-D tumors.
An average survival increase (as fraction of control) of 250% over monolayer was predicted
for drug-sensitive (MCF-7 WT) spheroids under various drug concentrations (Figure 5A),
while for drug-resistant (MCF-7 40F) spheroids the corresponding increase was 280% (Figure
5B). The median dose5 for MCF-7 40F was ~20% higher in spheroid vs. monolayer than for
MCF-WT, implying that not only did the 3-D morphology promote the net survival for both
cell types, but also that the resistant phenotype was further favored by the 3-D configuration.
We noted that drug-resistant spheroids maintained more compact, nearly spherical shapes
(Figure 6A) that simply decreased in size as cells were killed, while drug-sensitive spheroids
formed irregular, looser shapes (Figure 6B) that fragmented, particularly at higher drug
concentrations. The tumor morphology may depend on the competition between heterogeneous
cell proliferation caused by diffusion gradients of cell substrates, driving shape instability, and
stabilizing mechanical forces such as cell-cell and cell-matrix adhesion, as we quantitatively
showed in previous work [45]. Here, we observe a similar instability for drug-sensitive cells,
indicating that their adhesion is below the threshold necessary to maintain tumor shape stability.
This is confirmed through the computational model, where variation in the parameter values
for cell adhesion [22] reproduces these morphologies (Fig. 6C and D).
DISCUSSION
We have used an integrative methodology that tightly couples computational modeling with
biological data to quantify physiological resistance in 3-D tumor tissue in-vitro. The model
hypothesizes functional relationships that mechanistically link the growth and regression of a
tumor mass to the underlying phenotype. These relationships incorporate the complex interplay
between tumor growth, cell phenotype, and diffusion gradients, caused by heterogeneous
delivery of oxygen and cell substrates and removal of metabolites, and are thus able to calculate
tumor drug response as a predictable process dependent on biophysical laws. The results
underscore the importance of a quantitative approach in evaluating chemotherapeutic agents
that takes into consideration diffusion in addition to such chemo-protective effects as cell
phenotypic properties and cell-cell and cell-ECM (extracellular matrix) adhesion [49].
Although the model was calibrated with in-vitro data for breast cancer, this approach may also
apply to drug response of other tissues.
By modeling transport of drug and cell substrates, this work creates a quantitative tool that
could in the future be used to predict resistance in patients based on their tumor cell properties,
thus improving treatment outcome. In particular, the proposed functional relationships help to
quantify resistance in human breast cancer due to local cell substrate depletion. Our integrative
experimental/computational approach provides insight into the physical dynamics of solid
4256 nM exposure in-vitro for 96 hours yields an area-under-the-curve (AUC) of ~2.5×10−5 Molar.Hr, while for a typical patient Dox
plasma levels after a bolus injection represent an exposure of ~3×10−5 Molar.Hr.
5Median dose is the drug concentration required to achieve 50% cell kill.
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tumors, and validates the hypothesized functional relationships as adequate to describe the
observed phenomena. Although more complex functional forms could conceivably also
provide a reasonable match between model and experimental results, quantifying the diffusion
effect with a minimal mathematical description is the preferred approach, unless it is known
to be wrong, as it facilitates insight into the system and provides for more economic simulations.
We explicitly chose not to use complex relationships that would contain more parameters than
the number of independent measurements available for calibration, thus resulting in an
underdetermined problem.
Cell packing density affects the magnitude of diffusion gradients and may pose a barrier to
effective drug penetration [27]. This density can vary between cell lines and tumors and reflect
variations in drug resistance (cf. Figure 6). How this relates to cell adhesion molecule (CAM)
expression (e.g., integrin, E-cadherin) is unclear. Mechanisms of cell packing may include
stronger cell adhesion forces, due to higher E-cadherin expression; this should also limit
proliferation, which does not seem to be the case here for the drug-resistant cells at lower drug
concentrations. Additionally, cellular stress affects the quantity and strength of CAMs. We
have previously investigated the effect of the tumor microenvironment on tissue morphology
[50,45], suggesting that marginally stable environmental conditions could directly affect
morphogenesis and present an additional challenge to therapy [48,45] in-vivo by increasing
tumor cell invasiveness and leading to complex infiltrative morphologies, depending on the
magnitude of cell adhesion forces that tend to maintain compact, non-invasive tumors [50].
Diffusion gradients combined with higher cell packing density augment drug resistance
synergistically, as observed in our experiments with the higher median dose for the more
compact, drug-resistant spheroids, yet it may be difficult to deterministically gauge their
combined impact [5]. Synergism may be due to increased drug binding to ECM in tumor areas
proximal to the drug source, while substrate and drug penetration to distal areas is significantly
reduced due to higher cell packing, thus exacerbating the resistance effect of the diffusion
barrier.
Note that the presumption that the necrotic region is not a risk factor for progression may not
be necessarily true because selection pressures for resistance and induction of mutation are
strongest in hypoxic, peri-necrotic areas. Thus, inducing a large necrotic fraction during
treatment may paradoxically further select for drug resistance.
The class of prediction modeling presented in this paper, based on an assimilation of complex
processes with a fully 3-dimensional physical environment, offers the capability of
complementing current pharmacokinetic measurements. A more expansive integration of
theoretical model parameters with biological data could help to move towards prediction of
treatment response based on in-vitro and in-vivo tumor information, and determine the
correspondence between in-vitro measurements and the in-vivo condition to refine and validate
the assumption that this approach can describe in-vivo tumors. Incorporation of patient-specific
tumor phenotypic and microenvironmental parameters into the model could enhance clinical
strategies and prognosis evaluation. We further envision that these methods will enhance
current pharmacokinetic models used in designing and interpreting Phase II clinical trials.
Supplementary Material
Refer to Web version on PubMed Central for supplementary material.
ACKNOWLEDGEMENTS
We acknowledge John Sinek and Steven Wise (Mathematics, UC-Irvine), Sandeep Sanga (Biomedical-Engineering,
UT-Austin), Paul Macklin (SHIS, UT-Houston), Ernest Han (Obstetrics/Gynecology, UC-Irvine), Hoa Nguyen
Frieboes et al. Page 10
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(Medicine, UC-Irvine) for helpful comments and discussions, and the reviewers for their valuable input. Grant support:
The Cullen Trust of Health Care, NSF-DMS 0818104, National Cancer Institute, Department of Defense.
REFERENCES
1. Trédan O, Galmarini CM, Patel K, Tannock IF. Drug resistance and the solid tumor microenvironment.
J Nat Cancer Inst 2007;99:1441–1454. [PubMed: 17895480]
2. Lankelma J, Dekker H, Luque RF, et al. Doxorubicin gradients in human breast cancer. Clin Cancer
Res 1999;5:1703–1707. [PubMed: 10430072]
3. Sinek J, Frieboes HB, Zheng X, Cristini V. Two-dimensional Chemotherapy Simulations Demonstrate
Fundamental Transport and Tumor Response Limitations Involving Nanoparticles. Biomed Microdev
2004;6:297–309.
4. Primeau AJ, Rendon A, Hedley D, Lilge L, Tannock IF. The distribution of the anticancer drug
doxorubicin in relation to blood vessels in solid tumors. Clin Cancer Res 2005;11:8782–8788.
[PubMed: 16361566]
5. Sinek JP, Sanga S, Zheng X, Frieboes HB, Ferrari M, Cristini V. Predicting drug pharmacokinetics
and effect in vascularized tumors using computer simulation. J Math Biol 2009;58:485–510. [PubMed:
18781304]
6. Greijer AE, de Jong MC, Scheffer GL, Shvarts A, van Diest PJ, van der Wall E. Hypoxia-induced
acidification causes mitoxantrone resistance not mediated by drug transporters in human breast cancer
cells. Cellular Oncology 2005;27:43–49. [PubMed: 15750206]
7. Spitz DR, Sim JE, Ridnour LA, Galoforo SS, Lee YJ. Glucose deprivation-induced oxidative stress in
human tumor cells. Annals NY Acad Sci 2000;899:349–362.
8. Lee YJ, Galoforo SS, Berns CM, et al. Glucose deprivation-induced cytotoxicity and alterations in
mitogen-activated protein kinase activation are mediated by oxidative stress in multidrug-resistant
human breast carcinoma cells. J Biol Chem 1998;273:5294–5299. [PubMed: 9478987]
9. Brown NS, Bicknell R. Hypoxia and oxidative stress in breast cancer. Oxidative stress: its effects on
the growth, metastatic potential and response to therapy of breast cancer. Breast Can Res 2001;3:323–
327.
10. Li J, Lee AS. Stress induction of GRP78/BiP and its role in cancer. Curr Mol Med 2006;6:45–54.
[PubMed: 16472112]
11. Tomida A, Tsuruo T. Drug resistance mediated by cellular stress response to the microenvironment
of solid tumors. Anti-Cancer Drug Design 1999;14:169–177. [PubMed: 10405643]
12. Yun J, Tomida A, Nagata K, Tsuruo T. Glucose-regulated stresses confer resistance to VP-16 in
human cancer cells through a decreased expression of DNA topoisomerase II. Oncol Res 1995;7:583–
590. [PubMed: 8704275]
13. Pusztai L, Hortobagyi GN. High-dose chemotherapy: how resistant is breast cancer? Drug Resist
Updat 1998;1:62–72. [PubMed: 17092798]
14. Sanga S, Sinek JP, Frieboes HB, Ferrari M, Fruehauf JP, Cristini V. Mathematical modeling of cancer
progression and response to chemotherapy. Expert Rev Anticancer Ther 2006;6:1361–1376.
[PubMed: 17069522]
15. Sanga S, Frieboes HB, Zheng X, Bearer E, Cristini V. Predictive oncology: a review of
multidisciplinary, multi-scale in-silico modeling linking phenotype, morphology and growth.
Neuroimage 2007;37:S120–S134. [PubMed: 17629503]
16. El-Kareh AW, Secomb TW. Two-mechanism peak concentration model for cellular
pharmacodynamics of Doxorubicin. Neoplasia 2005;7:705–713. [PubMed: 16026650]
17. Ward JP, King JR. Mathematical modeling of drug transport in tumour multicell spheroids and
monolayer cultures. Math Biosci 2003;181:177–207. [PubMed: 12445761]
18. Jackson TL. Intracellular accumulation and mechanism of action of doxorubicin in a spatiotemporal
tumor model. J Theor Biol 2003;220:201–213. [PubMed: 12468292]
19. Norris ES, King JR, Byrne HM. Modelling the response of spatially structured tumours to
chemotherapy: Drug kinetics. Math Comp Model 2006;43:820–837.
20. Byrne HM, Owen MR, Alarcón T, Murphy J, Maini PK. Modelling the response of vascular tumours
to chemotherapy: a multiscale approach. Math Models Meth Appl Sci 2005;16:1219–1241.
Frieboes et al. Page 11
Cancer Res. Author manuscript; available in PMC 2010 May 15.
NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript
21. Panovska J, Byrne HM, Maini PK. A theoretical study of the response of vascular tumours to different
types of chemotherapy. Math Comp Model 2007;47:560–579.
22. Wise SM, Lowengrub JS, Frieboes HB, Cristini V. Nonlinear simulations of three-dimensional
multispecies tumor growth–I. Model and numerical method. J Theor Biol 2008;253:524–543.
[PubMed: 18485374]
23. Frieboes HB, Lowengrub JS, Wise S, et al. Computer simulation of glioma growth and morphology.
NeuroImage 2007;37:S59–S70. [PubMed: 17475515]
24. Zheng X, Wise S, Cristini V. Nonlinear simulation of tumor necrosis, neo-vascularization and tissue
invasion via an adaptive finite-element/level-set method. Bull Math Biol 2005;67:211–259.
[PubMed: 15710180]
25. Cristini V, Lowengrub J, Nie Q. Nonlinear simulation of tumor growth. J Math Biol 2003;46:191–
224. [PubMed: 12728333]
26. Sutherland RM. Cell and environment interactions in tumor microregions: the multicell spheroid
model. Science 1988;240:177–184. [PubMed: 2451290]
27. Grantab R, Sivananthan S, Tannock IF. The penetration of anticancer drugs through tumor tissue as
a function of cellular adhesion and packing density of tumor cells. Cancer Res 2006;66:1033–1039.
[PubMed: 16424039]
28. Fruehauf JP, Brem H, Brem S, et al. In vitro drug response and molecular markers associated with
drug resistance in malignant gliomas. Clin Cancer Res 2006;12:4523–4532. [PubMed: 16899598]
29. Fruehauf JP. In vitro assay-assisted treatment selection for women with breast or ovarian cancer.
Endocrine-Related Cancer 2002;9:171–182. [PubMed: 12237245]
30. Mehta RS, Bornstein R, Yu IR, et al. Breast cancer survival and in vitro tumor response in the Extreme
Drug Resistance Assay. Breast Cancer Res Treat 2001;66:225–237. [PubMed: 11510694]
31. Kern DH, Weisenthal LM. Highly specific prediction of antineoplastic drug resistance with an in
vitro assay using suprapharmacologic drug doses. J Nat Cancer Inst 1990;82:582–588. [PubMed:
2313735]
32. Roehm NW, Rodgers GH, Hatfield SM, Glasebrook AL. An improved colorimetric assay for cell
proliferation and viability utilizing the tetrazolium salt XTT. J Immunol Meth 1991;142:257–265.
33. Kim J, Kang K, Lowengrub J. Conservative multigrid methods for ternary Cahn-Hilliard systems.
Comm Math Sci 2004;2:53–77.
34. Jiang GS, Shu CW. Effcient implementation of weighted ENO schemes. J Comput Phys
1996;126:202–228.
35. Trottenberg, U.; Oosterlee, C.; Schüller, A. Multigrid. New York: Academic Press; 2001.
36. Isaacson, E.; Keller, H. Analysis of Numerical Methods. New York: Wiley; 1966.
37. Casciari JJ, Sotirchos SV, Sutherland RM. Glucose diffusivity in multicellular tumor spheroids.
Cancer Res 1988;48:3905–3909. [PubMed: 3383189]
38. Kallinowski F, Vaupel F, Runkel S, et al. Glucose uptake, lactate release, ketone body turnover,
metabolic micromilieu, and pH distributions in human breast cancer xenografts in nude rats. Cancer
Res 1988;48:7264–7272. [PubMed: 3191497]
39. Nugent LJ, Jain RK. Extravascular diffusion in normal and neoplastic tissues. Cancer Res
1984;44:238–244. [PubMed: 6197161]
40. Casciari JJ, Sotirchos SV, Sutherland RM. Variations in tumor cell growth rates and metabolism with
oxygen concentration, glucose concentration, and extracellular pH. J Cell Physiol 1992;151:386–
394. [PubMed: 1572910]
41. Teutsch HF, Goellner A, Mueller-Klieser W. Glucose levels and succinate and lactate dehydrogenase
activity in EMT6/Ro tumor spheroids. Eur J Cell Biol 1995;66:302–307. [PubMed: 7774614]
42. Acker H, Carlsson J, Mueller-Klieser W, Sutherland RW. Comparative pO2 measurements in cell
spheroids cultured with different techniques. Br J Cancer 1987;56:325–327. [PubMed: 3311111]
43. Durand RE. Slow penetration of anthracyclines into spheroids and tumors: a therapeutic advantage?
Cancer Chemother Pharmacol 1990;26:198–204. [PubMed: 2357767]
44. Kalra R, Jones A-M, Kirk J, Adams GE, Stratford IJ. The effect of hypoxia on acquired drug resistance
and response to epidermal growth factor in Chinese hamster lung fibroblasts and human breast-cancer
cells in vitro. Int. J Cancer 1993;54:650–655. [PubMed: 8514457]
Frieboes et al. Page 12
Cancer Res. Author manuscript; available in PMC 2010 May 15.
NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript
45. Frieboes HB, Zheng X, Sun C-H, Tromberg B, Gatenby R, Cristini V. An integrated computational/
experimental model of tumor invasion. Cancer Res 2006;66:1567–1604.
46. Freyer JP, Sutherland RM. Regulation of growth saturation and development of necrosis in EMT6/
Ro multicellular spheroids by the glucose and oxygen supply. Cancer Res 1986;46:3504–3512.
[PubMed: 3708582]
47. Carlsson J. A proliferation gradient in three-dimensional colonies of cultured human glioma cells.
Int J Cancer 1977;20:129–136. [PubMed: 903181]
48. Gatenby RA, Smallbone K, Maini PK, et al. Cellular adaptations to hypoxia and acidosis during
somatic evolution of breast cancer. Br J Cancer 2007;97:646–653. [PubMed: 17687336]
49. Morin PJ. Drug resistance and the microenvironment: nature and nurture. Drug Resist Updates
2003;6:169–172.
50. Cristini V, Frieboes HB, Gatenby R, Caserta S, Ferrari M, Sinek J. Morphologic instability and cancer
invasion. Clin Cancer Res 2005;11:6772–6779. [PubMed: 16203763]
Frieboes et al. Page 13
Cancer Res. Author manuscript; available in PMC 2010 May 15.
NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript
Figure 1. Computational model parameters
Functional relationships linking tumor mass growth and regression to the underlying phenotype
are based on these parameters, with values derived from experimental observations (obtained
previously and in this study). (*) Value shown is an example, corresponding to drug-sensitive
MCF-7 WT cells at 256 nM Dox.
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Figure 2. Diffusion gradients are incorporated in the model functional relationships to simulate the
experimental conditions
Clockwise, top left: Immunohistochemical staining of tumor spheroids showing increasing
positivity for pimonidazole protein adducts (darker brown) in the direction of the center,
indicating a decrease in oxygen across the viable rim of ~ 100 µm (v.r.: viable rim; n.: necrotic
region); NHE-1 staining showing up-regulation of Na+/H+ transporters (darker brown) towards
the necrotic region in response to a decrease in pH; increasing positivity for GLUT-1 staining
(brown) in the peri-necrotic region demonstrating cellular adaptation to hypoglycaemic and
hypoxic stress due to decreasing concentration of glucose and oxygen across the spheroid
radius; increasing positivity of TUNEL staining (dark brown) towards the center indicating
correlation of apoptosis with hypoxia; Ki-67 nuclear staining (dark gray) showing cell
proliferation correlating with substrate availability. Bars, 50 µm.
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Figure 3. Computational model computes the gradients in 3-D tissue
Cross-section of a tumor spheroid in the computational model (A) (dotted line: tumor boundary;
dashed: perinecrotic area) showing calculated diffusion gradients (arrow) of cell substrate
concentration σ/σ∞ (as fraction of external concentration in the medium) using diffusion
penetration length parameter L ≈ 100 µm. Necrosis becomes significant at substrate level σ/
σ∞ < 0.5. (B) Doxorubicin concentration d/d∞ in tumor tissue (normalized with concentration
in the medium) vs. distance from the tumor/medium interface, estimated for the model to be
consistent with data by [2,43], and others (Methods). By solving the unsteady Eq. (3), the drug
gradient across the viable region can be calculated at various times: bottom curve: 2 hours;
middle: 24 hours; topmost: steady state, showing ~ 2/3 drop from the tumor edge to the peri-
necrotic region.
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Figure 4. Cell proliferation and apoptosis
(A) Cell proliferation (ratio of proliferation-to-viability counts; n=3) in-vitro was lower by at
least 50% under conditions with diffusion gradients (spheroids) vs. monolayers for both cell
lines at drug concentrations d > 64 nM representing clinically relevant dosages. (B) Drug-
induced cell apoptosis death rate parameter (inverse time) vs. Doxorubicin
concentrations d calculated (Methods) from measured in-vitro monolayer viability counts
(n=3); glucose concentration σ∞ = 2.0 g/L.
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Figure 5. Validation of the hypothesized functional relationships
This is done by the computational model quantifying the physiological resistance introduced
by the diffusion gradients. Cell viabilities as fraction of control N/NC vs. Doxorubicin
concentrations d; glucose concentration σ∞ = 2.0 g/L; time T=96 hrs of drug exposure; n=3;
(A) MCF-7 WT drug-sensitive and (B) MCF-7 40F drug-resistant cells. White columns: 3-D
in-vitro tumor spheroids (dark gray: in-vitro monolayer data reported for comparison); light
gray: predictions of the computational model (error bars reflect variation in the apoptosis rate
parameter – see Methods).
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Figure 6. Correlation of tumor morphological stability with drug resistance
(A) Compact, nearly spherical morphology of a drug-resistant MCF-7 40F tumor spheroid in-
vitro is contrasted with the looser, irregular morphology of a drug-sensitive MCF-7 WT
spheroid (B). In the model, stable (C) and unstable (D) morphologies depend on variation in
the parameter values for cell adhesion forces [45]. Bar, 100 µm.
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