Computational
Modeling Identifies Morphologic Predictors of Tumor Invasion
Vittorio Cristini@,
Elaine Bearer, Robert Gatenby, Mauro Ferrari, Hermann B. Frieboes
VC: School of Health Information Sciences, The University of
Texas Health Science Center at Houston.
EB: Brown and Caltech. RG:
Arizona. MF: Houston. HF: Irvine
ABSTRACT
Tumor progression requires that the malignant cells acquire
the migratory phenotype and invade adjacent tissue. Yet the mechanisms driving
this invasion remain elusive. Here, we
derive a computational model that relies on the known characteristics of tumor
behavior and predicts the combination of variables most likely leading to
progression towards invasiveness.
Variables we considered include genotypic mutations in oncogenes, tumor
suppressors, and migration-associated genes; rates of proliferation, apoptosis
and in consequence nutrient consumption; nutrient concentrations and diffusion
rates; mobility function and velocity of tumor cells; and tumor mass effects.
As clones with higher rates of nutrient consumption arise within the tumor,
they create a landscape of differing nutrient concentrations. Computer simulations demonstrate that such
heterogeneity coupled with the other variables we considered is sufficient to
drive migration and proliferation of the emerging more aggressive clones up a
nutrient concentration gradient, both within and beyond the central tumor
mass. This substrate diffusion gradient
mediated by cell adhesion generates a morphologic instability at the tissue
scale that leads to collective assembly of tumor cells in clusters or
finger-like projections infiltrating into adjacent tissue and replacement of
less aggressive tumor cells within the tumor.
The morphology of computer-generated three-dimensional tumors is similar
to that of living glioma tumors in culture and ex vivo in human brain autopsy
specimens. By quantifying the link
between the (observable) morphology of the tumor boundary and the invasive
phenotype, we derive general principles of tumor cell behavior that point
towards new targets for intervention, such as improving nutrient supply.
Keywords:
Supplemental
data: a
detailed description of the mathematical model and numerical method that
produced the software used for the simulations presented here is at http://math.uci.edu/~cristini/publications/BMB
Part I.pdf and http://math.uci.edu/~cristini/publications/BMB
Part II.pdf.
INTRODUCTION
A wealth of qualitative empirical evidence links disease
progression with tumor morphology, pathology, invasion, and associated
molecular phenomena. Now what is needed is a quantitative method to determine precise functional relationships
between these parameters that would afford more accurate prediction of disease
progression from clinical observations of tumor morphology, e.g., patterns of
cell arrangements at the tumor boundary, and from analysis of specific genetic
mutations [44].
Our basic
hypothesis is that the characteristics of the tumor-host interface, e.g., its
“roughness” or harmonic content [29],
can be used to predict the underlying dynamical interactions between tumor cell
proliferation and adhesion, which, in turn, reflect both micro-environmental
factors influenced by cellular behavior. Using an integrative approach
combining mathematical models [32,33] with
in-vitro and in-vivo experiments [42] we
have previously demonstrated that morphological features at the tumor-host
interface can be used to predict underlying physiology. For example, these
studies demonstrated that local gradients in oxygen levels lead to reduction in
adhesion forces in hypoxic regions, causing instability in rapidly
proliferating areas and thus increasing invasive behavior with finger-like
tumor cell clusters at the periphery. Such invasive characteristics strongly
influence whether a tumor can be treated by local resection and limit other
treatment options as well.
Here we test our hypothesis in gliomas by
modeling the effects of oncogenes (that affect cell proliferation, motility and
nutrient consumption) and of tumor suppressor genes (that affect apoptosis and
motility) in a computational model that generates predicted tumor behavior.
Mutations in genes that regulate cell cycle and adhesion result in unrestrained
proliferation, invasion, and accumulation of further genetic damage
characteristic of high-grade disease
[1-8]. In particular, the oncogene EGFR is overexpressed, amplified or
mutated in glioma [9-13] and
promotes mitosis [7], tumor
progression in vivo [14], and
inhibits apoptosis [15]. The tumor
suppressor genes TP53 and Rb down-regulate cell division [7] and, hence, oxygen/nutrient consumption in glioma, while PTEN
controls angiogenesis, migration, and invasiveness [13,16-22]. These are inactivated in most malignant brain tumors [23-28].
Each oncogene/tumor suppressor combination will thus specify a particular
genotype in our in-silico idealized glioma model. Each mutation effects not
only the individual cell but also the tumor as a whole. As clusters of cells
arise with different local genotypes, proliferation and oxygen consumption
become progressively heterogeneous within the tumor mass, resulting in the
classic pleiomorphic appearance of tumors examined by histopathology.
We
introduce a biologically founded, multi-scale, mathematical model of tumor
progression in 3-D[1]
[31,65] to hypothesize quantitative functional relationships between the
effect of these genetic changes on tumor-scale cellular behavior, which affects
morphology and invasiveness of the tumor. We simulate tumor clones through mutations and selection and their complex
nonlinear two-way coupling with tumor mass growth, morphology, invasive phenotypes (i.e., cell chains, strands and
detached clusters [44]) and
development of necrosis in the presence of substrate
gradients in the microenvironment. We train this model using in-vitro and
in-vivo data.
We present
computer simulations, experiments and clinical data demonstrating that the
genetic evolution from lower- to higher-grade glioma and associated molecular
phenomena regulating cell proliferation, collective migration, and adhesion
forces generate, in a deterministic, quantifiable way, heterogeneous proliferation and oxygen/nutrient demand (and
suppression of apoptosis) across the three-dimensional tumor mass. The
associated effects on diffusion-gradients into and within the tumor lead to
local hypoxia, nutrient starvation and necrosis and to the formation, under low
cell adhesion, of wave-like patterns of cell rearrangements at the tumor
boundary and subsequent “morphologic instability” [29-33] with the development of these waves into strands, clusters
and individual cells proliferating and infiltrating from the tumor edge into
adjacent brain parenchyma. This supports our hypothesis by deterministically
connecting invasion to proliferation and nutrient heterogeneities produced by
underlying molecular phenomena in the cell-environment continuum. We correlate
these predictions by empirical examination of gliomas grown in culture and by
comparison of morphology with gliomas in human brain. These findings are supported by well-documented correlations of
invasion with hypoxia and necrosis induced in glioma by characteristically
abnormal, inadequate and hemorrhagic vasculature or by anti-angiogenic therapy [34-38].
METHODS
Multiscale model. The main
equations in the model are reported in Fig 1. Full details are available
in the Supplemental Data and in Ref. [31]. Also, the computer simulation
software is available upon request to the corresponding author. The array 
denotes lower-grade clone 1 and 
higher-grade clone 2. The rate of mutation lTR is a
biased random function 
of position and time within clone 1. More
malignant clones have evolutionary advantage in the model because they
proliferate more and die less, for example being resistant to hypoxia and
associated acidosis. The continuum-scale reaction-diffusion-advection equations
of the TUMOR model describing the local mass fractions of two tumor clones are
solved numerically either using finite-element/level-set [31] or
finite-difference/diffuse-interface methods1.
Necrotic and host tissues are described using analogous equations1 (not shown). A cell
adhesion energy1 enforces
phase separation of tumor and host tissues sharing a diffuse interface1 with thickness 1—100 mm. Additional cell adhesion at the tumor/tissue
interface, characteristic of collective cell migration [44], is set by an equivalent surface tension [43,31]. Cell velocity is determined through a modified Darcy’s law
[43] as a function of compressional
stress induced by proliferation in the tissue and of chemotaxis (and
haptotaxis) along substrate gradients; tumor
cells and extra-cellular matrix are treated as comprising porous media. The
hypoxia threshold is defined1
by the Heaviside function H. For
simplicity we only describe transport of one “nutrient” without distinction.
Notation “(·)+”
means “max{·, 0}”
because if the blood-to-tumor pressure difference approaches zero the blood
vessel will collapse [49].
Calculation of model parameters. Previous measurements of
growth and histology of in-vitro
ACBT (human glioblastoma multiforme) tumor spheroids [33], and of intra-cranial in-vivo BT4C (transformed fetal rat
brain) tumors in rats [50] were used. Briefly, higher-grade glioma (
)
parameters were set to 
and 
.
Diffusion penetration length L2=100 mm (
[29,33]), herein used as unit
length. Necrosis threshold nN/nV =0.5. Lower-grade glioma (
)
parameters were argued to be a fraction of the high-grade (based on their
relative time scales of growth [61,62]).
Consistently, the maximum mutation rate was set to 
.
A critical value of cell adhesion parameter was determined from shape stability
analysis of experimental and simulated spheroids [33]: compact spherical morphologies exist only for 
(after non-dimensionalization) [33]. Specific forms of cell adhesion
energy1, and values of
necrosis and tumor vascularization [31] parameters
were previously published. The above set of parameters provided the baseline
for our simulations. Each simulation corresponded to one fixed set of
parameters for clone 1 and another for clone 2 as described above. Parameter
sensitivity studies were performed where cell adhesion (
) and cell chemotaxis (
) parameters were varied to study their effect on the
morphology of infiltrating collective-cell patterns (i.e., cell chains vs.
strands vs. detached clusters [44]). Representative resulting
morphologies are reported in Figs. 3 and 4. The results
consistently confirmed that for relatively low cell adhesion, i.e. 
,
morphologic instability occurs when nutrient heterogeneity is present leading
to the development of such protrusions [33]. The shape features of these
protrusions further depended on the relative occurrence of cell proliferation
and cell chemotaxis. The control to all of these simulations was provided by
simulations corresponding to relatively high cell adhesion, i.e. 
(and no mutations),
for which tumors grow spherical and morphologic instability does not occur.
This critical value of cell adhesion is consistent to our prior model
calibration using morphologic instability of glioblastoma spheroids in vitro [33].
Correlation of histopathology of human glioma. As a first
step to correlate mathematical models with tumors in human specimens, four
archived autopsied brains obtained from the Brown University-Rhode Island
Hospital Brain Bank were examined in haemotoxylin-eosin stained paraffin
sections prepared according to standard autopsy procedures. Autopsied diagnosis
of glioblastoma multiforme was confirmed by two Neuropathologists, and
morphology at the tumor margins imaged on a Zeiss AxioImager by standard bright
field and by fluorescence using FITC and rhodamine filters. Selective
fluorescence in the rhodamine channel of hemoglobin in red blood cells combined
with autofluorescence of connective tissue in the FITC channel greatly enhances
detection of vasculature patterns in H&E sections of archived material.
RESULTS
The multiscale model (see Methods) developed in this
study considers genotype, phenotype, and tumor parameters (Fig. 1). The
genotype array M describes mutations regulating
cell proliferation, motility, nutrient/oxygen consumption, and apoptosis and
transforming clone 1 into clone 2 at a rate lTR (inverse
time). Phenotypical properties, e.g., mitosis, apoptosis, and necrosis rates,
cell adhesion and rate of cellular uptake of oxygen/nutrients are set from the
time- and space-dependent values of the components of M (see table embedded within Fig. 1).
We considered three general classes
of mutations: oncogenes, tumor suppressors and inactive genes. For simplicity,
we assumed that oncogenes double the mitosis rate and consequently the cellular
nutrient uptake; mutated tumor suppressor genes lower the cell death rate. The
model accounts for feed back from the microenvironment, i.e., mutations induced
by hypoxia [39-42], as the local
levels of oxygen/nutrients induce changes (Fig. 1, dashed) in the mutation function. At the tumor scale, the local
mass fractions of tumor clones, necrotic and host tissues are described.
Phenomenological parameters describe cell adhesion. Collective tumor cell
velocity depends on proliferation-driven mechanical pressure in the tissue,
chemotaxis and haptotaxis due to gradients of chemokines. Cell mass exchange
occurs due to mitosis, necrosis, apoptosis (and mutations of clone 1 into clone
2). Oxygen/nutrient availability limits the fraction of cycling cells. Regions
of tissue become hypoxic and then necrotic where nutrient/oxygen concentration
falls below a threshold. Death rates describe the disintegration of tumor cell
mass and the radial effusion of fluid away from the necrotic regions. Finally,
nutrient/oxygen delivery from neo-vasculature (via convection and diffusion [43,45,46]) and cellular uptake, and
nutrient/oxygen diffusion through the tumor tissue [47,48] are modeled. In this mathematical model, the effects of
each parameter on outcome can be tested individually or in combination.
The
onset of diffusion-driven shape instability [29,32,33] is illustrated in graphic output from our experiments [33] and simulations1 (Fig. 2).
Shape-destabilizing perturbations arise in the spatial arrangement of cells at
the outer rim of the tumor in spheroid bodies from human gliomas grown in
culture (Fig 2A). Our model replicates this type of asymmetric tumor
shape (Fig. 2B-D). Inhomogeneities in proliferation/chemotaxis gradients
(Fig. 2C) and in the local oxygen/nutrient concentration (Fig. 2D),
lead to spatially heterogeneous cell proliferation and migration as cells that
are exposed to more growth factors and chemokines proliferate more and move
faster. Mechanical forces, e.g., cell-cell and cell-matrix adhesion, are, in
general [44,33,29], stabilizing. In this case, cell
adhesion was not strong enough to prevent instability (
,
see Methods). Thus, when this set of
parameters is entered into the mathematical model, a typical tumor spheroid
resembling naturally occurring tumors is produced by the model.
When
instability persists, it leads to proliferative growth of “bumpy” protrusions
or clusters of cells (Fig. 2A,B, and Fig. 3). In vitro, these eventually detach as sub-spheroids from
the parent spheroid [33], analogous
to microsatellites in vivo, and may also represent the growth of chains,
strands or detached clusters of cells [44]
observed in tumors in vivo (Figs. 3B,D and 4C). Our model showed that strong adhesion can hamper the growth
of surface protrusions and enforce tumor spheroidal shape [33] as seen in tumors in vivo. Linear stability analysis reveals
that when proliferation is the prevailing pro-invasion mechanism, few, large
clusters form (long-wave-length perturbations [29]) as the four “bumps” in the example shown in Fig. 2B and those in Fig. 3C and eventually detach [33]. When chemotaxis or haptotaxis are
dominant, e.g., if mitosis is inhibited, protrusions begin as high-frequency
(short-wave-length [29])
perturbations on the tumor surface and develop into cell chains and strands [44] as shown in Fig. 3A. In all cases, these complex morphologic patterns developed
because cell adhesion parameters were set very low. Corresponding structures (Fig.
3B) observed after inducing hypoxia in spheroids in vitro and inhibiting
proliferation [53] and in xenografts
in vivo (Fig. 3D) [34] are
reported for comparison. Both these simulations corresponded to one single
clone (no mutations) and exhibited occurrence of morphologic instability
because this clone’s cell adhesion was
low (see Methods).
Computer
simulation of growth of a human glioma over time replicates tumor growth in
vivo (Fig. 4). For each time-snapshot in the computer simulation (Fig.
4A), the two-dimensional slices depict the spatial distribution of two
different clones. At very early
stages, expression of oncogenes and absence of tumor suppressor pathways result
in net growth of a relatively low-grade tumor. After about two months, the
tumor has expanded to approximately 3 mm by co-opting the brain vasculature
while retaining a compact shape with negligible necrosis. However, the
increased demand for nutrients has generated hypoxic gradients pointing
radially outwards from the lesion (these are shown at 6 months in Fig. 4B).
A second, more proliferative clone is generated by ongoing hypoxia-driven [39-42] mutations (see Methods) and grows. The higher cellular
uptake in clone 2 (indicated by arrows pointing to darker areas in tumor)
introduces perturbations in the spatial gradients of oxygen enhancing local
hypoxia (e.g., lower-left tumor corner). Hypoxic gradients generate spatially
heterogeneous cell proliferation and migration. After four months, this
perturbation has triggered a diffusional shape instability, which deforms the
tumor mass (cfr. onset of this mechanism illustrated by clusters of protruding
cells in Fig. 2A,B). Hypoxia and
necrosis are present within the regions where the more malignant clone is
growing. Shape instability leads to clusters of tumor cells of clone 2
protruding “finger-like” into the tumor mass of clone 1 first, and the host
brain later, growing at the expense of less proliferative clone 1 and of the
host tissue (we have also observed that detachment of these clusters may
occur). These fingers grow away from the bulk tumor in the direction of
substrate gradients.
Note that
other substrate components (e.g., glucose, growth factors, pH, build up of
metabolites both toxic and attractant) may be determinants of tumor morphology
in addition to or instead of oxygen, which was the only component considered in
the model for simplicity, by promoting genomic instability, proliferation, and
migration.
In about six months’ time, the
transition from low- to high-grade ensues at the entire lesion scale, while the
aggressive, invasive proliferation of clone 2 compromises tumor compactness.
This clone now proliferates and infiltrates in almost all regions of the tumor,
and in particular around the surface of the tumor. Hypoxic, necrotic areas
continue to expand. In eight months, the high-grade glioma infiltrates the
surrounding brain tissue. Clone 1 is being confined by competition with clone
2. Extensive necrosis is now present. In about ten months, invasive strands and
clusters are present. Clone 1 has been mostly eliminated from the area of tumor
invasion, and remains mostly stagnant. In a little over one year, the
surrounding brain tissue has been severely compromised by the high-grade
glioma. The expansion of clone 2, accompanied by continued necrosis, is now the
main determinant of tumor morphology.
A fluorescence image of
glioblastoma from one patient (see Methods) reveals a tip of an invading
finger (Fig. 4C) confirming the morphology of these infiltrative
clusters of cells predicted by simulation (e.g., Fig. 4A at 12 months).
These infiltrative shapes were consistently observed, although their size may
vary (not shown). Normal brain (white matter) to the left (Fig. 4C) has
fewer cell bodies and more abundant amorphous matrix. Invading malignant
astrocytes (right) have pleiomorphic nuclei and irregular distribution.
Engorged blood vessels within the tumor are visible as well as is a smaller
neovascular growth adjacent to the tumor boundary in the more normal appearing
brain parenchyma. According to the mathematical model, and supported by these
data, the cells may rely on vessels beyond the protrusions-and may grow towards
blood vessels that they stimulate. Older vessels within the tumor have thicker
walls (not shown) that are not as permeable for nutrient/oxygen exchange, thus
further promoting substrate gradients pointing outwards from the tumor mass.
Different tumors are likely to have different
genomic instability factors—different types and rates of mutations. The idealized tumor simulated here was
“programmed” to exhibit progressive appearance of one highly malignant clone
(clone 2). In reality, it is likely that multiple clones arise in time with
different degrees of malignancy. With this in mind, we reckon that the
simulated tumor size growth accurately resembled
the growth in size of human gliomas, i.e., it reached a size of ca. 4 cm in a
little over one year of simulated time.
The simulation illustrated in Fig.4
demonstrated that when tumors begin with less malignant, non-invasive clones,
following mutations are sufficient to modify the environment and trigger
invasion. Accordingly, other challenges are predicted by the model to lead to a
similar invasive outcome, such as if an area of the tumor is injured and
creates a nutrient imbalance without mutations or changing proliferation rate.
These predictions are consistent with the recent findings that anti-angiogenic
or hypoxia-inducing treatments trigger invasion (e.g., see Refs [34-36,53]).
DISCUSSION
Tumor size, shape, morphology, and invasive potential were
quantitatively predicted by incorporating information on the molecular scale
within a multiscale mathematical framework, in which morphology was modeled at
the continuum scale based on stability theory. Functional relationships across
genetic mutations, phenotype, and collective migration of structural cell
aggregates at the tumor scale [44]
were hypothesized. Experimentation within the mathematical framework allowed
variables such as proliferation rate, apoptosis, diffusional gradients and
clonal expansions to be tested.
Computer simulations captured disease progression from lower- to
higher-grade glioma. For simplicity[2],
single oncogene activation and tumor suppressor gene loss were modeled to
double cell proliferation and nutrient uptake and to decreased apoptosis.
Here we
are not stating that mutations in only two genes should be sufficient to induce
invasion. Instead, we are using computer modeling to isolate the effects of
individual mutations on tumor morphology. Over the course of one simulated
year, the higher-grade glioma co-opted most of the tumor mass, necrosis
developed, and as a consequence of these perturbations (exacerbation of
diffusion gradients) in the microenvironment, tumor morphologic instability was
triggered [32,33]. Fingering
protrusions of cell clusters were predicted to infiltrate surrounding tissue,
in agreement with our observations of glioblastoma in vitro [33] and in vivo. This infiltration was
driven mainly by the proliferation and collective migration of the more
aggressive clone, as has been observed in patient biopsies (e.g., [51,52] and this study). These results suggest that genetic mutations
that increase uptake of nutrient and augment cell proliferation (and indirectly
increase cell motility and reduce adhesion) have a quantifiable effect on
morphology at the tumor scale. In particular, they deterministically trigger invasive fingering into host tissue while
inducing necrosis (and angiogenesis) for certain parameter conditions. This
physical mechanism maximizes cell exposure to substrate by evading compact
nearly spherical morphology in favor of infiltrating “fingers”.
All known forms of collective cell migration (chains,
strands, detached clusters) [44] are
predicted by the stability theory (e.g., see [29]) for different
values of the molecular and environmental parameters. These infiltrative cell
clusters move in directions of increasing gradients of chemokine (oxygen, cell
nutrients) concentration. Inhomogeneous chemokine fields can be caused by
multi-clonality, abnormal angiogenesis or by therapeutic intervention such as
anti-angiogenic therapy, and destabilize a tumor through the mechanism
illustrated here [32,33,34-36].
Diffusional instability may provide a powerful tissue-invasion mechanism, since
it exposes tumor cells to sufficient substrate levels and thus allows them to
escape growth limitations imposed by diffusion and invade the host (even in
vitro [33,53]) independently of
the amount of angiogenesis [32,33].
Recent experiments with various glioma models in vivo [34-36,50,54] also support this hypothesis. For example, recently
published photographs of rat glioblastoma in vivo [50] showed that while the bulk tumor is perfused by blood, the
infiltrative cell clusters are much less perfused or not at all, thus
indicating that infiltration of the brain does not (directly) rely on
angiogenesis but on morphologic instability. These considerations apply to
tumor invasion beyond the context of glioma [44,55-58].
As demonstrated by our results, the interactions between
cellular proliferation and adhesion and other phenotypic properties are
reflected in both the surface characteristics of the tumor-host interface and
the invasive characteristics of the tumors. These cellular and molecular
properties are influenced by the cellular genetics and by micro-environmental
factors such as hypoxia. This allows observable properties of the tumor, such its
morphology in general and specifically the cell spatial arrangements at the
tumor boundary, to be used to both understand the underlying cellular
physiology and predict subsequent invasive behavior. Tumor morphology could be
used as a clinical prognostic factor not only because it would predict tumor
growth but also because it could indicate the presence of hypoxia and,
therefore, the potential for tumors to respond to oxygen-dependent treatments
such as radiation therapy and some chemotherapies. The mathematical models, in
turn, allow predictions of cellular and molecular perturbations that will alter
invasiveness and can be measured through changes in tumor morphology. This
opens the possibility of designing novel individualized therapeutic strategies in
which the microenvironment and cellular factors can be changed with the aim of
both changing tumor morphology and reducing tumor invasion. These methods could
be used to decrease invasiveness and promote defined tumor margins—an outcome
that would benefit cancer therapy by improving local tumor control through
surgery or radiation. In addition to
existing strategies that act on relevant cellular behaviors (for example
promotion of tumor cell adhesiveness [44]),
or that target oncogenes such as EGFR, tumor morphological stability could be
enhanced by “vascular normalization” [32,59],
or via uniform delivery of nanoparticles [60],
e.g., releasing oxygen and anti-angiogenic drugs or increasing pH, thus
enforcing a more homogeneous tumor microenvironment and normoxic conditions. By
maintaining microenvironmental homogeneity, the effects of genetic mutations
that lead to tumor morphological instability may be minimized, without direct intervention at the
genotype level.
ACKNOWLEDGEMENTS
We
acknowledge Steven Wise, Xiaoming Zheng and John Lowengrub (Mathematics, U. C.
Irvine) for help performing the computer simulations, Dr. Ed Stopa and the
Department of Pathology at Rhode Island Hospital for autopsied brain specimens,
Aleksey Novikov in ELB’s lab for technical assistance, and Henry Hirschberg
(Beckman Laser Institute, U. C. Irvine) for informing us of recent in vivo
results [50]. HF was supported by a grant from the National Science
Foundation (VC). ELB was partially supported by NINDS NS046810 (ELB), NIGMS GM47368 and by the Moore Foundation; VC also acknowledges the National
Cancer Institute for partial support.
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FIGURES
Figure 1. Multiscale mathematical model with hypothesized functional relationships describing the connection of molecular phenomena, phenotype and tissue growth (t is time; x is 3-D vector of spatial coordinates). The competing, morphology “destabilizing” and “stabilizing” factors [29,33] are highlighted (red). The most relevant values of the model parameters used are reported (table in PHENOTYPE box). Rates are inverse times (unit time=1 day).
Figure 2. Onset of
diffusional instability in glioblastoma (clone 2,
)
with collective motion of cells due to heterogeneous proliferation (and
possibly chemotaxis) up chemokine concentration gradients. In-vitro (A) spheroid (ca. 1 mm) (adapted from [33]) and computer simulation1
(B-D). Local mass fraction “C2” of
viable tumor cells (C) and oxygen
concentration “N” (D) (N=1 in the
medium). Note the hypoxic core (D)
and corresponding low levels of cell viability (C) within the spheroid, and the high viability at the outer rim.
Figure 3.
Persistence of instability over time. (A,C) Variability of morphologic
patterns predicted in silico [31] by
the mathematical model as a function of phenotype variables simulating in-vitro
(A) and in-vivo (C) conditions. Arrows: direction of time; red:
tumor boundary; black: hypoxia; blue and pink: neo-vascularization (C).
Morphologic instability observed in vitro (B) (adapted from [53])
and in vivo (D) (adapted from [34]).
Figure 4. (A)
Computer simulation of in-vivo human glioma (field of view=6 mm). Genotype 
(lower-grade clone 1) evolving to 
(higher-grade clone 2). Legend: local mass
fraction of clone 1 (“C1”) (clone 2: C2=1-C1). (B) Simulated
concentration of oxygen (“N” in legend) at 6 months, indicating hypoxic
gradients (N=1 in brain and lower in tumor). (C) Fluorescence image of
tumor front (right) pushing into more normal brain (field of view=0.5 mm). Note
green fluorescent outlines of vascular channels deeper in the tumor and clearly
demarcated margin between tumor and more normal brain to the left of the image.
Neo-vascularization at the tumor-brain interface is readily detected by the red
fluorescence from the red cells inside the vessels.