An integrated computational/experimental model of tumor drug response
Hermann B. Frieboes#,&, John P. Fruehauf$, Robert Gatenby+,@ and Vittorio Cristini#,X ,&,*
Departments of Biomedical Engineering#, Medicine—Hematology/Oncology$, and MathematicsX, University of California, Irvine.
Departments of Radiology and Applied Mathematics+, University of Arizona, Tucson.
Running title:
Computational/experimental model of drug response
Key words: computer
simulation; tumor drug response; tumor morphology; tumor spheroids;
mathematical model
ABSTRACT
The extent that
the cellular microenvironment influences tumor growth and drug
response in-vivo remains poorly understood. 3-D tissue architecture,
heterogeneous uptake of oxygen and cell substrates by solid tumors, caused by
unchecked growth, diminished apoptosis, treatment, or a combination of these,
establish diffusion gradients of these substances at the cellular
scale. Gradients may generate
marginally stable environmental conditions resulting in heterogeneous cell
proliferation, death, and nutrient uptake that may significantly affect tumor
mass increase and regression. In previous work, using a reaction-diffusion
mathematical model based on parameters directly describing tumor biology, we
showed that these gradients could influence tumor morphogenesis and raise a
further barrier to drug therapy by increasing tumor cell invasiveness and
supporting complex infiltrative morphologies depending on the strength of cell adhesion forces. Here we extend this work by creating a physiologically
based, integrated computational/experimental
model that hypothesizes a functional relationship directly linking solid tumor
mass increase and regression to the underlying phenotype through
cell-specific parameters modulated by diffusion gradients in the
microenvironment. We test this hypothesis by measuring parameters from
independent experiments and tasking the model with predicting the drug response
to cell cycle phase-specific chemotherapy. We show that the model corroborates
that substrate gradients introduce considerable resistance,
even under conditions of uniform drug distribution at cytotoxic levels, and accurately quantifies this
effect through the hypothesized functional relationship. The model also correctly predicts
that tissue compactness enhances resistance through steeper
diffusion gradients. These predictions are successfully tested against our in-vitro
experiments. This study suggests that in some cases
cell cycle phase-specific chemotherapy may be more effective in-vivo
when combined with treatment that minimizes diffusion gradients in the tumor
microenvironment, such as “vasculature
normalization” leading to reduction in nutrient
deprivation and hypoxia.
INTRODUCTION
Tumor resistance to chemotherapeutic drugs has been traced to molecular mechanisms that enhance individual cell survival, such as DNA damage repair, alterations in drug target, drug metabolism and cell cycle checkpoint mediators, decreased drug uptake, increased drug efflux, and diminished apoptosis (Dalton and Salmon, 1992, Gottesman et al., 2002). However, there is evidence that resistance beyond that intrinsic to and acquired by particular cells is introduced by the tumor mass itself as a three-dimensional physical object [11] affecting micro-scale conditions such as cellular arrangement and density, cell cycle stage [22], and DNA conformation [23]. The extracellular matrix may provide structurally based resistance [11], as could inter-cellular interactions, perhaps via adhesion molecules or tissue/cellular architecture [24, 22, 25] leading to apoptosis suppression through increased cell-cell adhesion [26]. Experiments studying cells in spheroids show that tumor mass-induced changes in protein expression [27], localization [28], and altered gene expression [29], such as mutation or amplification of genes encoding multidrug resistance protein efflux pumps, could favor drug resistance [10, 22, 30]. Drugs may be sequestered in outer tumor regions [31, 32], although active efflux by cells in inner layers may minimize tissue penetration barriers [30]. It is reasonable to assume that resistance introduced by tumor tissue in 3-D stems from substrate gradients in the cellular microenvironment causing hypoxia, hypoglycaemia, and acidosis (Fernandez et al., 2000), which may alter cell cycle kinetics [34], reduce cell proliferation [10], decrease drug sensitivity [21], foment a toxic cellular environment favoring upregulation of efflux mechanisms [30], and select for more resistant phenotypes (Raghunand et al., 2003).
Although intrinsic and acquired drug resistance at the individual cell scale may be enforced by molecular mechanisms, the above evidence strongly suggests that the tumor mass itself may select for, enhance, and trigger cellular-scale resistance. If the effects that a tumor mass exerts on drug resistance could be quantified based on specific tumor information, chemotherapeutic treatment would be substantially improved. In order to enable this quantification in a more rigorous manner, we created a three-dimensional, multi-scale computer (in-silico) model[1] that uses a mathematical formulation of tumor growth [8] and hypothesized a functional relationship linking mass increase and regression to the underlying phenotype through parameters describing cell mitosis, apoptosis and necrosis that are modulated by diffusion gradients. We measure values for these parameters from an independent set of experimental measurements, and then task the computer model with correctly predicting the drug response observed in-vitro. This model for the first time represents non-symmetric solid tumor growth in 3-D, thus providing the capability to capture the complexity and heterogeneity of the cellular microenvironment, including diffusion gradients of oxygen and cell substrates. We find that this integrated computational/experimental approach corroborates that a tumor mass introduces considerable resistance through gradients, and, more significantly, provides a means to quantify this effect through the hypothesized functional relationship.
Previous mathematical models, with
some important exceptions (e.g., Gatenby & Gawlinsky, 1996, Jackson & Byrne, 2000,
Gardner, 2000, Patel et al., 2001, Gatenby & Vincent, 2003, Zheng et al., 2005),
have yielded mostly qualitative results whenever experimental data (e.g., in-vitro drug
response [2, 4, 5] and in-vivo
drug delivery) were not used to
define and constrain model parameters but instead, equations were for the most
part fitted to the data, thus bearing no predictive value. In contrast, the integrated
computational/experimental methodology presented herein offers the possibility
of ultimately quantifying and predicting drug response under different
conditions in vivo through independent measurements of patient-specific
parameter values.
MATERIALS AND METHODS
In-silico model.
The fully three-dimensional mathematical model
of tumor growth uses a Cahn-Hilliard formulation1 of (Cristini: Update manuscript on
the web): mass fraction rH
of host tissue (or culture medium herein), rV
of viable and rD of dead tumor
tissue; tumor cell motion with local velocity u (and tissue pressure by Darcy’s law [13]).
Viable and dead tumor tissue equations are:
, (A1)
,
(A2)
where1 
is a positive cell mobility (assumed
constant), cell adhesion
forces are modeled1 using the variational derivative of the adhesion
energy E and a surface tension at the tumor/healthy tissue interface [13]. The “tumor” phase,
encompassing cells, interstitial fluid, and extracellular matrix, is treated as
porous media, and therefore no distinction between interstitial fluid
hydrostatic pressure and mechanical pressure due to cell-cell interactions is
made. Mass source terms SV and SD
quantitatively specify cell mass changes due to mitosis, necrosis, and
apoptosis, directly linking Eqs. (A) to the cell-scale, and include diffusion
of substrates such as cell nutrients and oxygen through tumor interstitium [14]. For simplicity, we describe transport of a generic substrate
of local concentration s within the
tumor:
, (B1)
, (B2)
where lM is rate of mitosis (rates are inverse times), lA is rate of apoptosis, lN is rate of necrosis, lL is rate of degradation of
necrotic tissue, sN is substrate limit for cell viability defining
the threshold for necrosis, and 
is maximum substrate value in the medium/tissue. The
Heaviside function H forces
the simulated1 interface between viable and necrotic tissue to be of
biologically realistic thickness (10–100 mm).
We hypothesize that substrate availability limits cell proliferation (e.g., the
glucose-regulated stress response can lead to cell-cycle arrest at the G1 phase
(Henning et al., 2004)
due to retention of growth factor receptors in the endoplasmic reticulum (Lee, 2001)) and, therefore, scale the
fraction of mitotic cells by the local fraction 
, enabling the modeling of substrate gradients in the microenvironment and quantification of
their effects on tumor mass increase and regression. These gradients are modeled1 across viable regions by
solving the reaction-diffusion equation
, (C)
where lengths are rescaled with a characteristic diffusion length, and
the negative term represents uptake by the tumor cells. The above system of
partial differential equations (A)-(C) is numerically solved in three spatial
dimensions and time using a finite-difference method1.
Cell culture.
MCF-7 WT (drug-sensitive) and MCF-7 40F (drug-resistant) breast cancer
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 37oC.
Tumor spheroids were formed by seeding 50,000 cells per well in 24-well Costar
3473 cytophobic plates (Corning, New York), shaking at 100 rpm for 10 min. on
day 1, and incubating for 3 days. On day 7, spheroids were fixed in 4%
paraformaldehyde and embedded in paraffin. 4 mm-thick cross-sections were stained with
H&E. Photographs were taken with a digital camera through a Zeiss
microscope (100x magnification).
Gradients in-vitro. Spheroid
samples were fixed in formalin and embedded in paraffin. To measure hypoxia, spheroids were incubated
in 200 μM pimonidazole in complete media for 2 hours at 37ºC. 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. (Gatenby: Need
to add antibody description for detection of GLUT-1 and Na+/H+
membrane transporters).
Drug response in-vitro.
Spheroids were exposed to Doxorubicin (Dox) (Bristol-Myers Squibb,
Princeton, New Jersey) concentrations ranging from 0 to 16384 nM in 4x nM
increments (0, 4, 16, 64, etc.) for approximately 96 hours. The following endpoints were concurrently
measured as fraction of negative control: proliferation using tritiated thymidine
(Amersham, Buckinghamshire, Great Britain) incorporation assay [15], viability using trypan
blue exclusion counts, and metabolic activity using XTT assay [16].
Drug response in-silico. Tumor spheroid growth was simulated in 3-D using the computer model1 briefly described above. Input parameters were calculated from independent measurements, thus fully and uniquely constraining the mathematical model. Necrosis parameters in Eqs. (B) for stable, diffusion-limited growth were calibrated by matching simulation results to in-vitro spheroid growth data, as described in detail previously [17]. These parameters are directly responsible for the steady spheroid size and extent of necrosis (after a period of growth) because they regulate the balance of volume growth due to cell proliferation in the viable region and volume loss due to cell disintegration in the necrotic center [17]. Briefly, based on our in-vitro measurements of spheroid and necrotic volumes we found a stable average spheroid mass of ca. 0.8 mm radius, with viable region of thickness ca. 0.1 mm (see also [18]). The model calibration based on these quantities consistently yielded: lN/lM=0.7 and sN/s¥=0.5.
Therapy was then applied in-silico to this simulated tumor spheroid for time T = 96 hours. Mass m created through proliferation and mass d lost due to drug uptake and subsequent apoptosis were calculated solving1 Eqs. (A) using Eqs. (B) at each drug concentration and were normalized by in-silico mass proliferation and total mass, respectively, during same timeframe for an untreated tumor spheroid, representing negative control:
, (D1)
, (D2)
where V is the tumor
spheroid volume at time t and lM,C is mitosis rate of negative
control (note that negative controls are stable spheroids with volume V(0)
corresponding to initial conditions of cytotoxicity experiments). Spheroid
viability reported in Results from computer simulations were calculated
as 
.
The rate lM (inverse time) measures the change in cell number in a population due to mitosis, normalized by total cell number, and thus was calibrated by matching proliferation data from our experiments and from simulations using Eq. (D1), using as initial guess for lM/lM,C, the in-vitro cell proliferation as fraction of control divided by in-vitro cell viability (total number of cells N) as fraction of control. Apoptosis rate lA (inverse time) was determined by a similar procedure using Eq. (D2) and initial guess (assuming first order kinetics) from in-vitro cell viability:
, (E)
where NC was in-vitro
viability of negative control, and the average substrate concentration was set 
by solving Eq.
(C) inside the spheroid volume V (this choice is corroborated by direct
experimental measurements, as described in Discussion).
RESULTS
Figure 1 (A) shows the cross-section produced by the computational/experimental model of a simple 3-D tumor spheroid growing in-silico prior to therapy, with values for mass density shown at the right. After the model parameter calibration described in Methods, computer simulations correctly predicted tumor mass growth and extent of necrosis in-vitro, also reproducing the stable three-dimensional spheroids observed in-vitro after a few weeks of growth. As previously observed (e.g., [10,17]), spheroids eventually stop growing as a result of a balance of cell proliferation in the viable region and cell necrosis in the center. Figure 1 (B) shows diffusion gradients predicted to develop across the tissue viable region in 3-D, leading to necrosis in the inner region, in agreement with previous experimental observations that substrate gradients are a leading cause of necrosis in MCF-7 spheroids (Spitz et al., 2000). Poor oxygenation in centers of spheroids in-vitro and tumor areas distal from capillaries in-vivo is believed to cause necrosis, although nutrient (e.g. glucose) concentration may affect necrosis development even when oxygen levels are significant [Spitz et al., 2000, 10, 38]. We note that hypoxia and hypoglycaemia may not necessarily be the direct cause of necrosis in spheroids since gradients of growth factors and expression of growth factor receptors may be just as important (Mueller-Klieser, 2000). For this simple case, the model predicts diffusion gradients to develop across a viable region of width ~ 0.1 mm and into a region of central necrosis. By solving Eq. (C), it can be shown that substrate concentration drops by at least 50% across the viable rim. A sample quantification of these gradients by the model is shown at the right of the figure.
Figure 1 (C) shows
upregulation of Na+/H+
transporters towards the necrotic region of MCF-7 spheroids in-vitro
in response to acidosis, corresponding to a gradient of pH across the viable
region. Hypoxia induces cells to switch
to anaerobic glycolysis, resulting in a more acidic environment, although
anaerobic glycolysis may also be observed under normoxic conditions (i.e.,
Warburg effect) (Gatenby
& Gawlinsky, 2003, Greijer et al., 2005). Figure 1 (D) shows
upregulation of GLUT-1 transporters in peri-necrotic regions of MCF-7 spheroids
in-vitro, corresponding to a gradient of glucose and further indicating
adaptation to hypoxia and hypoglycemia (Pascual et al., 2004). Figure 1 (E) shows
increasing numbers of hypoxic cells towards the necrotic region of MCF-7
spheroids in-vitro, indicating a gradient of oxygen. For example, an average oxygen concentration 
through tumors in-vitro
was previously reported [36], in excellent agreement
with our solution of the reaction-diffusion Eq. (C). According to our central
hypothesis in the model, these substrate gradients generated by the tumor mass
in 3-D have a direct effect on cell-scale parameters (mitosis, apoptosis,
necrosis) and are responsible for the increased resistance of three-dimensional
tissue (e.g., spheroids) over that of individual cells.
Diffusion gradients observed with spheroids in-vitro are similar to those occurring in-vivo [35, 36], and they may enforce drug resistance at the cellular-scale in several ways. Hypoxia, acidosis, and hypoglycaemia in the tumor microenvironment can lead to accumulation of unfolded or malfolded proteins in the endoplasmic reticulum (ER), triggering transcription of several genes as part of the unfolded protein response: GRP78 (Lee, 2005, Park et al., 2004), whose expression has been linked to Dox resistance in MCF-7 cells (Dong et al., 2005), GRP94, whose elevated basal levels have been detected in MCF-7 cells with reduced proliferative capacity (Gazit et al., 1999), and HSP27, which has been shown to cause resistance to Dox in MCF-7 cells (Fuqua et al., 1994). Dox cytotoxicity for MCF-7 cells is not affected by hypoxia (Kalra et al., 1993, Greijer et al., 2005), but rather is mainly due to gradients of glucose (Tomida & Tsuruo, 1999). Glucose starvation causes the glucose-regulated stress response (Lee, 1987), which has been correlated with resistance to topoisomerase II-directed chemotherapeutic drugs such as Dox (Tomida & Tsuruo, 1999, Fernandez et al., 2000, Li & Lee, 2006) through a decreased expression of topoisomerase II (Shen et al., 1989, Yun et al., 1995), as observed in MCF-7 cells (Yun et al., 1995). Glucose deprivation further induces cellular oxidative stress in MCF-7 cells (Lee et al., 1998, Spitz et al., 2000). Persistent oxidative stress at sublethal levels, caused by endogenous mechanisms in breast tissue, exposure to Dox, and glucose deprivation, promotes MCF-7 cell viability and resistance to apoptosis (Brown & Bicknell, 2001). MCF-7 resistance to Dox is also abetted by gradients of growth factors (e.g. serum), reducing cell proliferation (Lee et al., 1997, Wosikowski et al., 1997, Hug et al., 1986) and leading to proliferation gradients (Zhang et al., 2005) with outer cells having highest mitotic activity and cells near necrotic regions having reduced proliferative rates or being mostly quiescent [39, 40]. Cell cycle arrest minimizes cytotoxicity because Dox induces antineoplastic and toxic effects through DNA intercalation and inhibition of DNA and RNA polymerases [44], DNA alkylation [45], as well as interaction with topoisomerase II [46]. Drug gradients might induce further resistance [47] due to physical barriers associated with tissue morphology that Dox molecules must overcome to reach their nuclear target [31, 10]. Acidosis due to gradients of pH positively charges Dox, decreasing its passive translocation over the cellular membrane (Mahoney et al., 2003, Kozin et al., 2001), and by ion gradients on drug distribution and ion trapping, where Dox, as a weakly basic drug, will concentrate in more acidic cellular compartments instead of reaching its intracellular target (Raghunand et al., 2000). Acidosis reduces uptake of Dox in MCF-7 cells (Greijer et al., 2005). In addition, the effect of Dox on topoisomerase II activity is optimal at alkaline pH (Gieseler et al., 1996).
Figure 2 shows results from independent set of experiments performed to determine values for input parameters of the integrated computational/experimental model, as described in Methods. In-vitro responses to Dox were measured in spheroids and monolayer with respect to cell viability (A), proliferation (B), and metabolic activity (C) endpoints. In the control, MCF-7 40F cells were about 5 times more viable than MCF-WT in monolayer, and about equal in spheroid (data not shown). This could be due to a higher necrosis rate for MCF-40F cells in 3-D, which in our culture conditions were observed to proliferate faster (~3x) than MCF-7 WT cells, as well as to enhanced survival of MCF-7 WT cells in spheroids.
As expected, MCF-7 40F cells exhibited higher resistance to Dox than
MCF-7 WT cells (within experimental error). The resistance differential was not
affected by drug concentration either in monolayer or in spheroid. Drug sensitivity of MCF-7 WT cells
may result from an acidification defect within vesicles of the recycling and
secretory pathways as a consequence of an inability to protonate, sequester,
and then secrete Dox (Schindler
et al., 1996). A
decrease in Dox accumulation has been observed in MCF-7 resistant cells, with
no change in subcellular distribution and no increase in accumulation in
vesicles (Lee et al., 1997). Additionally, MCF-7 40F resistance to Dox
could be based on efflux mechanisms previously observed in MCF-7 drug-resistant
cells (Lee et al., 1997),
although we verified through immunohistochemistry that the MCF-7 40F cells did
not express Pgp (MDR1).
More MCF-7 WT cells were killed in monolayer than MCF-7 40F cells, whereas in spheroids the number of net surviving cells was similar for both (data not shown), indicating that 3-D tissue dramatically increased resistance for MCF-7 WT. In fact, spheroids had higher viability than monolayers by a similar amount for both MCF-WT and MCF-40F cells, exhibiting a resistance differential of up to 45x. This matches well with MCF-7 resistance up to 50x observed previously to another topoisomerase II-directed drug (etoposide) by induction of stress conditions in monolayer replicating glucose deprivation (Yun et al., 1995), pointing to the role of substrate gradients in generating resistance in 3-D tissue.
The median dose for MCF-7 40F was ~20% higher in spheroid vs. monolayer
than for MCF-WT (Figure 2(A)), 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. However, individual cell
proliferation (ratio of proliferation-to-viability counts, Figure 2 (B)
to (A)) and metabolic activity (ratio of metabolic-to-viability counts, Figure
2 (C) to (A)) were the same or higher in monolayers than in spheroids
for both cell lines at almost all drug concentrations, indicating that a 3-D
tissue conformation was less favorable for cell proliferation and metabolism,
and again pointing to the role of substrate gradients. Slower MCF-7 cell proliferation in spheroids
vs. monolayer has been previously observed (diFaute et al., 2002).
Cytotoxicity depended on cell counts in viable regions and was found to
be independent of overall
spheroid shape during therapy both in-vitro and in-silico. Final in-silico
viable region volumes as fraction of negative control matched in-vitro
viabilities for both cell lines. For
example, in-silico therapy at a drug concentration of 16384 nM yielded a
viability ratio of 0.35 in agreement with in-vitro. We observed that drug-resistant spheroids (Figure
3 (C)) maintained more compact (diFaute et al., 2002), nearly spherical shapes that simply shrunk as
cells were killed, while drug-sensitive spheroids (Figure 3 (D))
developed irregular, looser shapes, suggesting that compactness enhanced tissue mass-driven resistance [31] by enforcing steeper diffusion gradients. Compactness (tumor cell density) can
vary between different cell lines, reflecting variations in drug
resistance, and may be enhanced by drug exposure [24]. Mechanisms
of compactness may include stronger cell-cell and cell-matrix adhesion forces,
such as higher E-cadherin expression inhibiting apoptosis [50]. In our simulations, higher cell adhesion parameter values1
were used to simulate resistant spheroids than for sensitive ones, following a
recently presented calibration procedure [17]. The model predicted the same difference in
compactness (Figure 3, (A) and (B)). We have investigated this
previously in detail [17,20]
using experiments and simulations.
Figure 4 shows high correlation (0.97 ≤ R2
≤ 0.99) between the in-vitro drug response (cell viability for
MCF-7 WT (A) and MCF-7 40F (B) spheroids), and that predicted by
the computer model from Eqs. (D) as described in Methods. Cell
proliferation predicted by the model for spheroids of both cell lines yielded
the same high degree of correlation (Figure 4 (C) and (D), 0.93
≤ R2 ≤ 0.99). In agreement with in-vitro
results, therapy in-silico yielded increasingly larger tumor mass
regression as drug concentration increased.
DISCUSSION
We formulated a physiologically based computational/experimental model that hypothesizes
a functional relationship linking tumor mass increase and regression (Eqs. A in
Methods) to the underlying cell phenotype via parameters describing cell
mitosis, apoptosis, and necrosis (Eqs. B).
This functional relationship provides a means to quantify the effects of
tumor-mass induced diffusion gradients in 3-D on drug resistance (Figure 1 and Eq. C).
Input parameter values for the model were set from independent experiments (Figure
2 and Eq. E), and the model was tasked with predicting tumor cell viability
and proliferation under various drug concentrations (Figure 4 and Eqs.
D). We confirmed that tumor tissue
introduces considerable resistance to cell cycle phase-specific
chemotherapy through substrate
gradients in 3-D, and correctly and accurately quantified this effect through
the hypothesized functional relationship by achieving a high correlation
between the experimental and the computer model predicted drug responses.
In
previous work we hypothesized that local hypoxic conditions or spatial hypoxic
and substrate gradients in 3-D, such as during tumor growth and anti-angiogenic
therapy, may create marginally stable environmental conditions. These could directly affect tumor
morphogenesis and present an additional barrier to drug therapy [17, 51] by increasing
invasive capability of tumor cells and leading to complex infiltrative tumor
morphologies depending on the magnitude of cell-cell adhesion forces [20]
that tend to maintain compact noninvasive tumors. Here we provide quantitative evidence that these tumor
mass-induced diffusion gradients could in some cases be the primary component
of cell cycle phase-specific drug resistance in-vivo beyond the
resistance originating from intrinsic and acquired factors such as cell
genotype (e.g. loss of p53) and phenotype (e.g. efflux pumps) by downgrading cell proliferation. Cell
viability in 3-D tissue culture in our experiments (Figure 2) remained
relatively high at higher drug dosages while cell proliferation was severely
diminished (see also [52]),
and individual cell
metabolic activity was generally lower for cells in spheroids than in
monolayer. If tumor cells act
like normal resting cells, for which effects of a drug such as Dox are
minimized, they might,
for example, be able to survive hypoxic conditions for up to 10 days in-vivo
[53], primarily through
downregulation of mitochondrial function [54], and resume proliferation once micro-environmental conditions become more
auspicious. Our results suggest that consistently
minimizing diffusion gradients in the 3-D tumor microenvironment could help
maintain drug sensitivity even in the presence of cell-scale genotypic and
phenotypic characteristics that oppose it.
This objective could be achieved by supporting a properly working tumor
microvasculature (e.g. “vascular normalization” [56]) and by minimizing heterogeneity in this
environment during therapy. Strategies
focused solely on bulk destruction of tumor mass through periodic chemotherapy
and reduction of vascular density via anti-angiogenic therapy, although perhaps
augmenting patient survival in the short term, may paradoxically increase
long-term drug resistance.
This work further supports previous experimental [11,12,24] and theoretical [57] findings suggesting that
three-dimensional tumor morphology may significantly affect treatment response
through diffusion gradients and tissue compactness. We are currently evaluating how this is influenced by
intercellular adhesion and stromal matrix gene expression across different cell
lines. Recent studies suggest that
tumorigenesis is favored when the extra-cellular matrix becomes chronically
stiffer or integrin-ERK-Rho activity remains elevated for long times [58]. Using these types of data, future computer
simulations of response can be performed that take into account the complex in-vivo
drug delivery dynamics, for which vascular density (e.g. CD31) is an
important variable. The expansion of
our model to represent in-vivo drug response would allow quantification
of factors such as dosage, concentration, number and duration of cycles, and
drug types to be used in combination, enabling drug
therapy optimization on a patient-specific basis. To assess the predictive capabilities of this methodology,
model results can be compared to in-vivo DC MRI data of patients’ tumor
response to therapy [59].
ACKNOWLEDGEMENTS
We gratefully acknowledge John P. Sinek and Steven M. Wise (Mathematics, U.C. Irvine), Ernest Han (Obstetrics and Gynecology, U.C. Irvine Medical Center), and Hoa P. Nguyen (Medicine, U.C. Irvine Medical Center) for helpful comments and discussions. We thank the collaboration of Felicity Rose (Pharmacy, U. of Nottingham) with the immunohistochemistry experiments.
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FIGURE
LEGENDS
Figure 1. (A): Cross-section produced by the calibrated computational/experimental model of a simple 3-D tumor spheroid growing in-silico prior to therapy. Tumor boundary is denoted by the dotted line, with values for the mass density shown on the right. (B): Diffusion gradients (from normal values to below normal, following the arrow from tail to head) are hypothesized to develop in the 3-D tissue, through a viable region of width ~ 0.1 mm and into a region of necrosis (significant at substrate level s/s¥ < 0.5). Values on the right indicate a sample quantification of these gradients by the model. Necrotic area is enclosed by the dashed line. Bottom: Histological cross-sections of MCF-7 tumor spheroids showing substrate gradients observed through the viable region in-vitro. (C): Upregulation of Na+/H+ transporters towards the necrotic region indicates a gradient of pH. (D): Upregulation of GLUT-1 membrane transporters in the peri-necrotic region indicates a gradient of glucose. (E): Hypoxic cells indicate a gradient of oxygen. Symbols: s.: substrate; v.r.: viable region; n.: necrosis. Bar, 100 mm.
Figure 2. Independent set
of experiments performed to determine values for input parameters of the
integrated computational/experimental model. (see Methods). (A): Cell viability in monolayer and
spheroids. (B): Cell metabolic
activity in monolayer and spheroids. (C):
Cell proliferation in monolayer and spheroids.
Figure 3. Prediction by the integrated
computational/experimental model of tumor morphologies as a function of
tissue compactness. Top left: Spheroid in-silico with high tissue
compactness is nearly spherical. Top
right: Spheroid in-silico with low tissue compactness has unstable
morphology. Bottom left: Observation of spheroid in-vitro with
high tissue compactness (MCF-7 40F). Bottom
right: Observation of spheroid in-vitro with low tissue compactness
(MCF-7 WT).
Figure 4. Predictions by the integrated
computational/experimental model of spheroid viability ((A) and (B))
and proliferation ((C) and (D)).
This is accomplished by hypothesizing a functional relationship
linking tumor mass increase and regression to the underlying phenotype through
cell-specific parameters modulated by tumor mass-induced diffusion gradients in
the microenvironment (see Methods). For comparison, we show traditional, non-predictive pharmacodynamic
Hill dose response curves that are simply fitted to the data via 
(a, b are
fitting parameters, q is drug concentration).
FIGURES
Figure 1.
Hermann B. Frieboes
Figure 2. Hermann B. Frieboes
Figure 3. Hermann B. Frieboes
Figure
4. Hermann B. Frieboes