Unformatted text preview: Research Article Acid-Mediated Tumor Invasion: a Multidisciplinary Study
1,2 4 1,3 Robert A. Gatenby, Edward T. Gawlinski, Arthur F. Gmitro,
Brant Kaylor, and Robert J. Gillies Departments of 1Radiology, 2Applied Mathematics, and 3Optical Sciences, University of Arizona, Tucson Arizona;
and 4Department of Physics, Temple University, Philadelphia, Pennsylvania Abstract
The acid-mediated tumor invasion hypothesis proposes
altered glucose metabolism and increased glucose uptake,
observed in the vast majority of clinical cancers by fluorodeoxyglucose-positron emission tomography, are critical for
development of the invasive phenotype. In this model,
increased acid production due to altered glucose metabolism
serves as a key intermediate by producing H+ flow along
concentration gradients into adjacent normal tissue. This
chronic exposure of peritumoral normal tissue to an acidic
microenvironment produces toxicity by: (a ) normal cell death
caused by the collapse of the transmembrane H+ gradient
inducing necrosis or apoptosis and (b ) extracellular matrix
degradation through the release of cathepsin B and other
proteolytic enzymes. Tumor cells evolve resistance to acidinduced toxicity during carcinogenesis, allowing them to
survive and proliferate in low pH microenvironments. This
permits them to invade the damaged adjacent normal tissue
despite the acid gradients. Here, we describe theoretical and
empirical evidence for acid-mediated invasion. In silico
simulations using mathematical models provide testable
predictions concerning the morphology and cellular and
extracellular dynamics at the tumor-host interface. In vivo
experiments confirm the presence of peritumoral acid
gradients as well as cellular toxicity and extracellular matrix
degradation in the normal tissue exposed to the acidic
microenvironment. The acid-mediated tumor invasion model
provides a simple mechanism linking altered glucose metabolism with the ability of tumor cells to form invasive cancers.
(Cancer Res 2006; 66(10): 5216-23) Introduction
Fluorodeoxyglucose-positron emission tomography imaging has
shown that the vast majority of human cancers exhibit significantly
increased glucose flux compared with normal tissue (1, 2). This
property seems to be a characteristic of invasive neoplasms and can
be used to distinguish benign from malignant lung nodules (3).
Increased glucose uptake is observed coincident with the transition
from colon adenomas to invasive cancer (4) and from carcinoma
in situ to invasive breast cancer (5). Furthermore, several studies have
shown that increasing glucose uptake correlates with increasing
tumor aggressiveness and progressively poorer prognosis (6–8).
The observed increase in glucose demand occurs on top of
mitochondrial energy production and reflects an unregulated increase in the consumption and trapping of glucose beyond the
cells’ ability to oxidize pyruvate (9, 10). Some of the elevated glycolysis likely reflects adaptive changes to regions of intratumoral
hypoxia that are caused by disordered vascularization with temporal
and spatial variations in blood flow and oxygen delivery (11, 12).
However, constitutive up-regulation of glycolysis is also observed
even in the presence of adequate oxygen supplies (aerobic glycolysis): a phenomenon first noted by Warburg >80 years ago (13, 14).
We propose that this altered glucose metabolism and flux in
malignant cells plays a critical role in cancer biology (15–17).
Briefly, we hypothesize that the glycolytic phenotype first emerges
as a survival mechanism in the regions of intermittent hypoxia that
occur in premalignant lesions (17). These hypoxic regions are
established as hyperplasia increases the spatial separation between
intraluminal cells and their blood supply, which remains in the
stroma separated from the tumor cells by an intact basement
membrane. These dynamics result in cycles of hypoxia-normoxia
(18). Adaptation to this unstable environment includes constitutive
up-regulation of glycolysis, which remains elevated even in the
presence of oxygen (in anticipation of the next anoxic episode).
Elevated glycolysis also results in greater acid production, which is
exacerbated by the increasing distance between cells and the acid
sink provided by the blood vessels. Microenvironmental acidosis
could lead to cellular necrosis and apoptosis (19, 20), adding
additional selection forces that drive cancer cells to evolve
phenotypes with increased resistance to acid-induced cellular
toxicity (21, 22).
The net result of this evolutionary sequence is a cellular
phenotype with a powerful adaptive advantage. These ‘‘aggressive’’
cancer cells alter their microenvironment by increased production
of glycolytically derived acid. This is toxic to normal cell
competitors but less harmful to the cancer cells themselves.
An extension of this concept is the acid-mediated tumor
invasion hypothesis (15, 16, 23). We propose that invasive cancers
continue to use the glycolytic phenotype to their advantage, thus
explaining the persistence of aerobic glycolysis in clinically evident
primary cancers and metastasis. The model includes the following
components: Requests for reprints: Robert A. Gatenby, Department of Radiology, University
Medical Center, 1501 North Campbell Avenue, Tucson, AZ 85718. Phone: 520-626-5725;
Fax: 520-626-9981; E-mail: [email protected]
I2006 American Association for Cancer Research.
doi:10.1158/0008-5472.CAN-05-4193 Cancer Res 2006; 66: (10). May 15, 2006 5216 1. Increased glycolysis of cancers alters the microenvironment
by substantially reducing intratumoral pHe—a phenomenon
observed experimentally (24–26).
2. H+ ions produced by the tumor diffuse along concentration
gradients into adjacent normal tissues probably carried by a
3. Acidification of the extracellular peritumoral environment is
advantageous to the tumor because it:
. induces normal cell death due to necrosis or caspasemediated activation of p53-dependent apoptosis pathways (19, 20) and death of normal cells produces potential
space into which the tumor cells may proliferate; www.aacrjournals.org Acid-Mediated Tumor Invasion
. . . extracellular acidosis also promotes angiogenesis through
acid-induced release of vascular endothelial growth factor
and interleukin-8 (27, 28);
acidosis indirectly promotes extracellular matrix degradation by inducing adjacent normal cells ( fibroblasts and
macrophages) to release proteolytic enzymes such as
cathepsin B (29), or increased lysosomal recycling (30);
acidosis inhibits immune response to tumor antigens (31). As discussed above, we propose that, during carcinogenesis,
tumor cells evolve a phenotype which is adapted to environmental
acidosis and is resistant to acid-mediated toxicity. This is observed
experimentally as tumor cells survive and proliferate in pHe
significantly lower than that of normal cells (21, 22, 32), perhaps
due to constitutive up-regulation of H+ transporters or mutations
in p53, caspase, or downstream effectors (22). In vivo , this
phenotype confers a significant growth advantage as tumor cells
proliferate in the acidic environment of the tumor-host interface
allowing them to invade into the damaged normal tissue. Thus, the
tumor edge can be envisioned as a traveling wave extending into
normal tissue following a parallel traveling wave of increased
microenvironmental acidity (16).
In the current report, this proposed mechanism of tumor
invasion is tested in silico using mathematical models. We then
present experimental observations of a peritumoral pHe gradient
extending into normal tissue and evidence of acid-induced toxicity
in normal cells—both critical predictions of the hypothesis and of
the mathematical models. Materials and Methods maximum value in empty space and going to zero when cells are closely
packed), then Eq. A becomes
þ r Á DN1 1 À
¼ r1 N1 1 À
¼ r2 N2 1 À
þ r Á DN2 1 À
K1 K2 ðBÞ where D N 1 and D N 2 (in cm2/s) are the ‘‘empty-space’’ diffusion constant of
the normal and tumor cells, respectively. For simplicity, we will assume that
these are approximately equal: D N 1 % D N 2 = D N . The Lotka-Volterra terms
ensure that the density-dependent diffusion parameters are always positivedefinite 2[0, D N ].
Building the model: effects of local pH. Next, we assume that each cell
type has an optimal pH for survival and that if the local pH is perturbed
from that optimal value, in either an acidic or an alkaline direction, the cells
begin to die. We also assume that the death rate saturates at some
maximum value when the environment is extremely acidic or alkaline. The
simplest ad hoc functional form meeting these criteria is an ‘‘inverted
H À H1;2
f1;2 ðH Þ ¼ d1;2 1 À exp À
where H is the local concentration of H+ ions (in mol/L), d 1,2 are the
saturated death rates (in 1/s), H 1,2 are the local H+ ion concentrations
(in mol/L) corresponding to the optimal pH’s, and H 1,2
are the halfwidths of the inverted Gaussian (in mol/L). Including the death rates
Eq. D into Eq. B, we finally get
À f1 ðH ÞN1 þ DN r Á 1 À
¼ r1 N1 1 À
K1 K2 Mathematical Model
The tumor-host interface is a highly complex system dominated by
nonlinear processes. The dynamics of this class of systems typically exhibit
nonintuitive properties including extreme sensitivity to critical parameter
values and rapid transitions between steady states with discontinuities and
bifurcations. For this reason, we initially explored the hypothesis with
mathematical models to test its feasibility in silico and gain some initial
understanding of expected system dynamics (see Fig. 1). Below, we outline
our general approach. Mathematically inclined readers are encouraged to
review Appendix A for more details.
Building the model: spatial constraint of growth and migration. The
acid-mediated tumor invasion hypothesis can be framed mathematically as
a system of reaction-diffusion equations.
Using an approach previously described (16), we begin with spatial
constraint: if N 1 and N 2 denote the cell densities (in cells/cm3) of the
normal and tumor populations and assuming these populations only
compete for available space, then their temporal evolution is governed by
the following equations ðDÞ
¼ r2 N2 1 À
À f2 ðH ÞN2 þ DN r Á 1 À
¼ r1 N1 1 À
¼ r2 N2 1 À
where r 1,2 and K 1,2 are the growth rates (in 1/s) and spatial carrying
capacities (in cells/cm3) of the respective populations. If it is also assumed
that cells can migrate through space via a process akin to Fickian diffusion,
in which the diffusion variables are themselves density-dependent (having a www.aacrjournals.org Figure 1. The dynamics of the tumor-host interface predicted by simulations
from the mathematical model. The tumor edge is a traveling wave moving left to
right preceded by a wave of acid extending into the peritumoral normal tissue.
This results in a complementary traveling wave of receding normal tissue moving
left to right as a result of acid-induced toxicity. 5217 Cancer Res 2006; 66: (10). May 15, 2006 Cancer Research
Building the model: acid production and uptake. We assume that
H+ ions are produced at a rate proportional to the local concentration of
tumor and removed by the combined effects of buffering and vascular
evacuation, both of which are proportional to microvessel areal density.
¼ r3 N2 À d3 ðH À H0 Þ þ D3 r2 H
@t ðE Þ where H is the H+ ion concentration (in mol/cm3), r 3 is the H+ ion production
rate (in mol/(cell s)), d 3 is the H+ ion uptake rate (in 1/s), H 0 is the H+ ion
concentration in serum, and D 3 is the H+ ion diffusion constant (cm2/s). Experimental Methods
Tumors. In vivo experiments were done using two cell lines: MCF7/s and
PC3N/enhanced green fluorescent protein (eGFP). The former is a human
breast cancer that grows relatively slowly in vivo , whereas the latter is a
rapidly growing human prostate cancer. Prior in vitro studies had measured
proliferation rates, acid production, acid tolerance, and acid diffusion rates in
both cell populations. Both lines were transfected with GFP to allow accurate
tumor size and edge detection in vivo using fluorescent microscopy (Fig. 2).
Experiments were done in severe combined immunodeficiency (SCID)
mice (6-8 weeks of age; 25-30 g) bred and housed in a defined flora animal
colony. A dorsal skin fold chamber (Fig. 2) was surgically implanted under
anesthesia (75 mg of ketamine and 25 mg of xylazine per kg s.c.), as
described previously (33). After a 2-day recovery period, the coverslip in the
chamber was gently lifted and a slurry of 2.5 to 3 Â 106 tumor cells were
placed on the exposed surface near the center of the chamber. Tumor
growth was monitored using fluorescent microscopy approximately every 2
days. pHe experiments were done when tumors reached a diameter of f3.0
mm. Subsequent pHe imaging was determined by tumor growth as assessed
by fluorescent microscopy. The PC3N/eGFP typically began to grow
immediately following placement so that images were generally obtained
every 2 to 3 days. The MCF7/s tumors exhibited a long lag phase in which
there was initially no growth so that pHe maps were typically obtained every
5 to 7 days. Imaging continued until the tumor occupied >50% of the
chamber area. In some cases, the tumors did not grow and imaging was
discontinued once tumor regression was observed.
pHe measurements. Extracellular pH was measured using SNARF-1
(Molecular Probes, Eugene, OR), which exhibits a spectral shift in
fluorescence emission with change of pHe and has been well described in
the literature (34). Spatial distribution of pHe in the tumor and adjacent
normal tissue was obtained using ratiometric imaging with two sets of data
measuring the intensity values collected in two different spectral emission
regions and converting the ratios to a pH image using calibration data.
Images were obtained with a Nikon Eclipse E-600 microscope with a
Nikon C-1 confocal microscope attachment in epi-illumination mode. Light
sources on this instrument include two helium:neon lasers at 543 and 632
nm, and an argon laser operating at 488 nm. Fluorescence detection was
obtained through three photomultiplier tubes (PMT) set to detect
fluorescence emission using a 515/30 nm filter, a 595/50 nm filter and a
640 nm long pass filter, respectively. Channel 1 was used to view the
emission from GFP; channels 2 and 3 were used to capture the two spectral
signals from the SNARF fluorescence emission. The 543 He:Ne laser was
used to excite the SNARF fluorescence and the argon laser was used to
excite the GFP. The signals from the PMTs are read by Nikon’s EZ-C1
software and displayed as an image. The software is capable of
simultaneously collecting 12-bit images from each PMT channel.
During the imaging procedure, the mice were anaesthetized with
ketamine HCl (100 mg/mL), xylazine (20 mg/mL), and acepromazine
maleate (10 mg/mL) (Phoenix Pharmaceuticals, Inc., Belmont, CA). The
anaesthetized mouse was placed in a Plexiglas holder and the window
chamber attached rigidly to the microscope stage to prevent movement.
Initially, a GFP image was captured with both the 2Â and 1Â objective to
accurately determine the tumor borders. Fluorescent images were then
obtained using both the 2Â and 1Â objectives and the 543 He:Ne laser to Cancer Res 2006; 66: (10). May 15, 2006 Figure 2. A, the dorsal window chamber in a SCID mouse. B, a fluorescent
micrograph (original magnification, Â2) showing GFP-transfected tumor cells
allowing definite identification of the site of the tumor-host interface. obtain background fluorescence to be subtracted from subsequent images.
Two hundred microliters of the 1 mmol/L SNARF solution in saline was
injected via the tail vein catheter. Images were then collected using the green
He:Ne laser with both objectives at 15, 30, and 40 minutes after injection.
The autofluorescence background image taken before injection of the
dye was subtracted on a pixel-by-pixel basis from the SNARF fluorescence
images to obtain only the SNARF fluorescent signal for each channel. The
background-subtracted images were smoothed (i.e., convolved with a 2 Â 2
rect function) before calculating the ratio image. This smoothing has the
effect of reducing high-frequency noise, but spatial resolution is also
reduced. The images from each channel were smoothed before the ratio was
calculated. The intensity ratios were converted to pHe images following
calibration for SNARF-1 in buffered solutions of varying pH as measured by
a pH electrode in a 96-well plate. Three sets of calibration data were taken
from the same solution on consecutive days.
Analysis of pHe gradients at the tumor-tissue interface was accomplished
via the following procedure. The centroid and peripheral edge of the tumor
were determined from the high-contrast GFP image. The pHe image was
then segmented into eight directions defined as angular segments of 45
degrees from the centroid of the tumor. Within each angular segment, a
binary image of tumor versus nontumor was created based on the tumor
edge. Binary dilation and contraction operations were employed to grow or
contract the edge of the tumor by selected distances in steps equivalent to a
distance of five pixels. These dilations and contractions defined tissue
regions extending either out or in, respectively, from the tumor edge. The
pHe values were then averaged in these regions to yield the average pHe
relative to distance from the edge. The values of pHe were then plotted as a
function of distance from the edge for each of the eight angular segments.
In some cases, angular segments were discarded if they did not correspond
to valid data (e.g., the tumor was at the edge of the field of view in the
window chamber so that for certain angular segments there was no 5218 www.aacrjournals.org Acid-Mediated Tumor Invasion
‘‘normal’’ tissue outside the tumor boundary). The pHe data can be used to
estimate the flow of H+ ions at the tumor edge. This is done after binning
the 512 Â 512 image to 64 Â 64. The gradient at the centroid of four
adjacent points is calculated. The gradient array is displayed as arrows, a
built-in capability of the IDL program. The arrows are then overlayed on the
preexisting ratio image.
Microscopic evaluation. After completion of the sequence of pHe
imaging experiments, the xenograft tissues were harvested, fixed in 10%
neutral buffered formalin for 24 hours, processed and embedded in paraffin.
Routine H&E and periodic acid Schiff (PAS) stains were done on 3 Am
sections of tissue. Cleaved caspase-3 was detected by immunohistochemistry using the Ventana Medical Systems (Tucson, AZ) Discovery XT automated platform. Rabbit polyclonal anticleaved caspase-3 (Cell Signaling
Technology, Danvers, MA) was incubated for 2 hours at 37jC at a dilution of
1:200, and detected with a biotinylated streptavidin-horseradish peroxidase
and 3,3¶-diaminobenzidine detection system. Results
Model results. Numerical simulations from the models can then
be done using parameter estimates based on experimentally
determined proliferation rates, acid production, and acid-induced
toxicity in the cell lines used in subsequent experiments. This
allows the models to produce detailed predictions about cellular
and microenvironmental dynamics at the tumor-host interface.
The details of this analysis are included in Appendix A.
In Fig. 1, numerical solutions show that the interface, at any
given time, represents a snapshot of a traveling wave as tumor cells
advance and normal cells recede. The tumor wave is preceded by a
gradient of excess H+ extending into adjacent normal tissue. Within
the region of peritumoral acidosis, the models predict a loss of
normal tissue due to acid-induced cellular toxicity and extracellular
matrix breakdown. These results support the feasibility of the acidmediated invasion model. The models showed that we should be
able to experimentally detect a peritumoral acid gradient. Using
parameter estimates available in the literature, it seemed likely that
observation of the gradient and associated toxicity would require
a spatial resolution in the range of V50 Am. This limited the
appropriate experimental approach to fluorescent microscopy
rather than, for example, magnetic resonance imaging or positron
Experimental results. The calibration studies showed that pH
resolution of the fluorescent images was, on average, 0.042 pH units
and none of the pH values varied by >0.02 over 3 days.
In vivo measurements showed that all of the tumors exhibited a
significantly acidic pHe when compared with normal tissue.
Average pH values across the entire regions of interest in growing
tumors were 6.91 F 0.14 for MCF-7 tumors (n = 4) and 6.83 F 0.21
for the PC3N tumors (n = 10). These data compare favorably to
pHe values measured using other approaches. For example,
although the average pHe decreased with tumor size, the pHe of
small (3-500 mm3) MCF-7 tumors was 6.99 F 0.11 as measured
with 31P MRS (35).
All of the PC3N/eGFP tumors showed a gradient of acidification
extending from the tumor edge into the adjacent tumor over a
typical distance of 100 to 400 Am on the first postimplantation
images. All but one of these tumors continued to exhibit a significant gradient during subsequent imaging. As shown in Fig. 3, the
gradient in the initial images was typically quite uniform but less so
on later imaging. This spatial heterogeneity seemed to be the result
of angiogenesis because the tumor was relatively avascular on the
first images, but showed significant vascular growth in later
studies. The expected flow of H+ from the tumor edge into the www.aacrjournals.org peritumoral normal tissue as a result of the gradients is shown
in Fig. 4.
One PC3N/eGFP tumor failed to maintain a pHe gradient into
adjacent normal tissue and also exhibited no growth before finally
A peritumoral pHe gradient was not observed initially in the
MCF7/s tumors which also did not exhibit any significant growth
for f21 days following implantation. However, following this lag
phase, rapid growth was observed simultaneously with onset of
complete acidification of the chamber. Because all of the normal
tissue in the chamber became acidic, the depth of the peritumoral
gradient could not be determined but was clearly larger than that
of the prostate cell line.
Three of the MCF7/s tumors, despite successful initial implantation, failed to grow and eventually involuted. In all cases, the
initially acidic intratumoral pHe returned to normal values as
tumor growth failed.
The relatively shallow peritumoral pHe gradient observed in the
PC3N/eGFP tumors allowed the local effects on the normal tissue
within the acidic gradient to be assessed. Following the in vivo Figure 3. pHe gradients at the PC3N/eGFP tumor-host interface along radians
drawn from the tumor center. The tumor-host interface is designated as the
0 point on the x -axis. All of the experiments showed a peritumoral acid gradient
that was qualitatively and quantitatively similar to the results from the
mathematical model in Fig. 1. Values obtained 2 days following placement of the
tumor slurry (A ). The relatively avascular tumor shows a fairly uniform pHe
distribution and gradient. Values obtained 4 days later (B ). During that time,
significant tumor growth was observed. Note that the pHe distribution is less
uniform, presumably representing increasing microenvironmental heterogeneity
due to variations in tumor vascular distribution and flow. 5219 Cancer Res 2006; 66: (10). May 15, 2006 Cancer Research provide insights into the governing nonlinear dynamics not
obtainable intuitively. These models show the feasibility of acidmediated tumor invasion and made detailed predictions regarding
the cellular and microenvironmental dynamics of the tumor-host
interface which could be tested experimentally (Fig. 1).
The in vivo experiments presented in this study confirm the
modeling predictions that tumors acidify the extracellular space of
normal tissue around the tumor edge. This gradient of acidosis
seems quite variable in size ranging from 100 to 400 Am in the
PC3N/eGFP line and at least a few millimeters in the MCF7/s line.
Our observations suggest that this heterogeneity in the acid
gradients is likely dependent on variations in vascular density and
blood flow. Each blood vessel may act as a H+ sink depending
on flow rate and the acid gradient across the vessel wall. On
histologic sections, we found a significantly increased (but highly
variable) vascular density in the tumor edge and the normal
tissue immediately adjacent to the edge (see Fig. 6, for
example). Integrating the interactions of vessel growth, blood Figure 4. A map of peritumoral H+ flow using vectors generated from the pHe
distribution around PC3N/eGFP. The tumor is the darker region (left ) and the
tumor-host interface is drawn based on the GFP image. Arrows, direction of H+
flow and the length of each arrow is dependent on the slope of the gradient
(the steeper the gradient, the longer the arrow). Note the general flow of H+ from
the tumor core to its periphery, and from there, into the normal tissue,
although there is significant heterogeneity. experiments, the mice were sacrificed and the tissue in the chamber
was removed and fixed. As shown in Fig. 5, caspase stains from these
samples showed evidence of apoptosis in multiple cells adjacent to
the tumor edge in the acidic regions shown on FRIM images. H&E
stains also show evidence of toxicity in skeletal muscle cells
immediately adjacent to the tumor edge but not those more distant.
In Fig. 6, PAS stains showed evidence of considerable degradation in
the extracellular matrix immediately adjacent to the tumor edge. Discussion
The acid-mediated tumor invasion hypothesis proposes that
increased glycolysis, a phenotypic trait almost invariably observed
in human cancers, confers a selective growth advantage on
transformed cells because it allows them to create an environment
toxic to competitors but relatively harmless to themselves.
Specifically, this model hypothesizes that cancer cells use
inefficient glycolytic pathways even in the presence of oxygen
because it results in increased acid production, and a decrease in
microenvironmental pHe. Through an evolutionary sequence
during carcinogenesis, tumor cells evolve phenotypic adaptations
to the toxic effects of acidosis including, for example, increased H+
transport against concentration gradients across the cell membrane and mutations in acid-induced apoptotic pathways. Normal
tissue, lacking these adaptive traits, is vulnerable to acid-mediated
toxicity including cell necrosis and apoptosis, and degradation of
the extracellular matrix by acid-induces release of cathepsin B and
other proteolytic enzymes.
This proposed mechanism of tumor invasion is initially
evaluated through mathematical models. Because the tumor-host
interface is a highly complex structure, mathematical modeling can Cancer Res 2006; 66: (10). May 15, 2006 Figure 5. Photomicrographs of the PC3N/eGFP-host interface following H&E
and caspase staining. Apoptotic cells are present (dark area ) are present within
and adjacent to the tumor edge (small arrow ). Skeletal muscle immediately
adjacent to the tumor edge (large arrowheads ) shows evidence of toxicity
with loss of normal striation, swelling, and increased eosinophilia. Skeletal
muscles more distant from the tumor edge (dashed arrows ) retain a normal
appearance. This generally corresponds to the depth of the pHe gradient
observed on FRIM images. 5220 www.aacrjournals.org Acid-Mediated Tumor Invasion and extracellular matrix degradation in this acidic region supporting,
but not confirming, the proposal that the gradient plays an
important role in promoting tumor invasion. This suggests that
continued investigation is warranted both to increase understanding
of the critical intracellular and extracellular interactions at the
tumor-host interface and develop novel tumor therapy strategies
based on perturbations of those system dynamics (23, 36). Appendix A. Nondimensionalization of the Model
Equations D and E can be nondimensionalized using the
g1 ¼ N1 =K1
g2 ¼ N2 =K2
K ¼ H =H0 ðF Þ s ¼ r1 t n¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
r1 =DN x which transforms Eqs. D and E into
¼ g1 ð1 À g1 À g2 Þ À /1 ðKÞg1 þ rn Á ½ð1 À g1 À g2 Þrn g1
@s Figure 6. PAS staining of thePC3N/eGFP-host interface regions. There is
clear loss of peritumoral extracellular matrix in the immediate region (dashed
arrows ) of the tumor edge (thin arrows ) roughly corresponding to the acidosis
gradient. The more distant extracellular matrix remains more intact (larger
arrows ). flow, microenvironmental properties, tumor growth, and acidinduced toxicity is a major future goal of this work.
In the PC3N/eGFP line, the peritumoral acid gradient was
confined to a region of the chamber and, thus, the potential cellular
and extracellular effects of the gradients could be assessed. We did
observe evidence of cellular toxicity, apoptosis, and extracellular
matrix degradation in the peritumoral normal tissue roughly
corresponding to the acid gradient. These findings are consistent
with the predictions of the model although limited by their
observational and nonquantitative characteristics. Furthermore, we
cannot unambiguously determine cause and effect so that it is
possible, for example, that the peritumoral pHe gradient and tissue
toxicity may represent manifestations of another underlying
process such as increased tumor interstitial pressure. This is
particularly the case for observed toxicity in skeletal muscle, which
is relatively tolerant of acidic environments (at least over short
periods of time). Clearly, additional studies will be required to
definitively evaluate the acid-mediated tumor invasion hypothesis.
In conclusion, our multidisciplinary study shows the potential
benefits of combining mathematical modeling with experimental
studies in the investigation of complex systems dominated by
nonlinear dynamics such as the tumor-host interface. Our results
show the presence of a peritumoral acid gradient in two xenograft
models—confirming a critical prediction of the acid-mediated tumor
invasion hypothesis. Our results show evidence of cellular toxicity www.aacrjournals.org @ g2
¼ q2 g1 ð1 À g1 À g2 Þ À /2 ðKÞg2 þ rn Á
½ð1 À g1 À g2 Þrn g2 ðG Þ @K
¼ q3 g2 À d3 ðK À 1Þ þ D3 r2 K
where q 2 = r 2/r 1, q 3 = r 3K 2/(H 0r 1), d 3 = d 3/r 1 and D3 = D 3/DN .
The death rate functions are also dimensionless having the form
" ( /1;2 ðKÞ ¼ d1;2 1 À exp À opt K À K1;2 width
2K1;2 ! 2 )#
ðH Þ opt
= H 1,2 /H 0
where d 1,2 = d 1,2/r 1, Ã1,2 = H 1,2 /H 0, and Ã1,2
are all dimensionless as well. Appendix B. Parameter Estimation
In vitro spheroid doubling times are between 1 and 4 days,
therefore, we take r 2 = ln 2/2.5 days % 3.2 Â 10À6/s. For normal
tissue wound healing, 4 days seems reasonable for the doubling
time, therefore, we take r 1 = ln 2/4.0 days % 2.0 Â 10À6/s. We
assume that the volume limited carrying capacities of tumor and
normal tissue are the same: K 1 = K 2 % 5 Â 108 cells/cm3. 5221 Cancer Res 2006; 66: (10). May 15, 2006 Cancer Research For vascular evacuation without buffering d 3 = ap , where a %
200/cm is the vessel areal density and P % 1.2 Â 10À4 cm/s is the
vessel permeability for lactate resulting in a removal rate of 2.4 Â
10À2/s. Local buffering might increase this by 25%, thus, our final
estimate for this rate is d 3 % 3.0 Â 10À2/s.
If we assume the serum pH0 = 7.4 is also the optimal pH for
normal tissue growth, we have H opt = H 0 = 3.98 Â 10À11 mol/cm3.
An optimal pH of 6.8 for tumor growth gives H opt = 1.58 Â 10À10
The acid production rate is trickier to estimate, therefore, we
work backwards from known data. Assuming that we have a tumor
sufficiently large that the temporal and spatial derivatives at its
core are small. From Eq. E, we see that r 3 % d 3 (H core À H 0) / K 2.
Assuming a core pH of 6.4, we get r 3 % 2.2 Â 10À20 mol/cell s.5
The lactic acid and cellular diffusion constants are, respectively,
D 3 % 5 Â 10À6 cm2/s and DN % 2 Â 10À10 cm2/s.
The dimensionless variables are, using the above values, as
q 2 = r2/r1 Appendix C. Fixed Points
The fixed points of the model and their stability must be
determined numerically. To find the fixed points, we use the acid
equation to eliminate Ã from the cellular equations (i.e., the first
two in equations with the derivatives set to zero). This leaves two
nonlinear equations in two unknowns, i.e., g 1 and g 2. Fortunately,
we know that these are physically bounded on the interval (0,1).
Therefore, we partition the domain (0 V g 1 V 1) B (0 V g 2 V 1) into
a fine rectangular grid with Dg 1 = Dg 2 = 0.01 and use the grid
positions as starting points for a multidimensional NewtonRaphson algorithm. The value of Ã corresponding to a NewtonRaphson solution for the cellular densities is found using the third
equation. We save the unique solutions (i.e., those that do not differ
from others as determined by the condition j gðiÞ À gðjÞ j Ve , with
the components of g being g 1 and g 2, and q = 1 Â 10À3) and
determine their stability by numerically computing the eigenvalues
of the full three-dimensional Jacobian.
The fixed point analysis for the variables given in the table
above is: 1.6 q 3 = r3K2/(H0r1) 1.4 Â 105 d 3 = d3/r1 1.5 Â 104 D3 = D3/DN 2.5 Â 104 Ãopt = Hopt/H0
1 1.0 Ãopt = Hopt/H0
2 4.0 Ãwidth = Hwidth/H0
1 0.1 Ãwidth = Hwidth/H0
2 0.4 d 1 = d1/r1 2.0 d 2 = d2/r1 2.0 A plot of the equation versus Ã using the last six variables is
shown in the following figure: Fixed point no. 1 is trivial and nos. 4, 5, and 6 are nonphysical.
Fixed point no. 2 has the tumor beating the normal, however, it is
unstable. That leaves fixed points no. 2 and no. 7: although both are
stable, the tumor will propagate into the normal because it is in
some sense ‘‘more stable’’ (note the relative magnitudes of the
corresponding eigenvalues). Appendix D. Dynamic Simulation
We have developed a method-of-lines parabolic solver that can
be used to solve equations subject to the fixed point boundary
conditions determined as described above. Initial conditions are
taken to be step functions that connect the left and right fixed
point boundary conditions.
The following is a typical screen output produced by the solver.
<c1>, <c2>, and <c3> are the velocities of the normal, tumor, and
acid wavefronts. 5
This value is remarkably consistent with the curve fit results of the Martin and
Jain (34) data to our original, more simplistic model (20). Cancer Res 2006; 66: (10). May 15, 2006 5222 www.aacrjournals.org Acid-Mediated Tumor Invasion The wavefront velocities are in dimensionless form and must be
multiplied by the velocity scale factor r1 DN .
In Fig. 1, we show the profiles after 600 time steps. Notice the
interesting features on the tumor and acid edges which correspond
to the point in space at which the acid level is optimal for the
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