J. Biomedical Science and Engineering, 2010, 3, 501-508 JBiSE
doi: 10.4236/jbise.2010.35070 Published Online May 2010 (http://www.SciRP.org/journal/jbise/).
Published Online May 2010 in SciRes. http://www.scirp.org/journal/jbise
Diffusive modelling of glioma evolution: a review
Alexandros Roniotis1,2, Kostas Marias1, Vangelis Sakkalis1, Michalis Zervakis2
1Foundation for Research and Technology-Hellas, Institute of Computer Science, Heraklion, Greece;
2Technical University of Crete, Department of Electronic and Computer Engineering, Chania, Greece.
Received 26 January 2010; revised 27 January 2010; accepted 24 February 2010.
ABSTRACT
Gliomas, the most aggressive form of brain cancer,
are known for their widespread invasion into the tis-
sue near the tumor lesion. Exponential models, which
have been widely used in other types of cancers, can-
not be used for the simulation of tumor growth, due
to the diffusive behavior of glioma. Diffusive models
that have been proposed in the last two decades seem
to better approximate the expansion of gliomas. This
paper covers the history of glioma diffusive model-
ling, starting from the simplified initial model in 90s
and describing how this have been enriched to take
into account heterogenous brain tissue, anisotropic
migration of glioma cells and adjustable proliferation
rates. Especially, adjustable proliferation rates are
very important for modelling therapy plans and per-
sonalising therapy to different patients.
Keywords: Glioma; Brain Tumor; Diffusive Models;
Proliferation; Invasion
1. INTRODUCTION
Glioma is a type of cancer of central nervous system that
starts in the brain or in the spine. It is called a glioma
because it arises from glial cells. The most common site
of gliomas is the brain. Gliomas constitute more than
50% of all brain cancer cases. Despite extended research
in this area, patients are rarely given more than 12
months survival time [1].
The diagnosis of gliomas can be done by MRI, CT,
angiogram or biopsy, with MRI being the most common
method. After glioma is diagnosed, treatment is directly
necessary. However, the most important problem with
diagnosis and treatment is that gliomas are characterized
by invasiveness. This means that there are cells diffused
beyond the imaged tumor, which cannot be visualized by
common imaging techniques. Thus, even if the clinician
allows some safety margin during resection around the
imaged tumor, cancer is expected to recur.
These dispiriting results have forced researchers
worldwide to work with understanding the glioma grow-
th procedure and its special pathology. One of the rapid-
ly emerging fields in glioma study is tumour growth
modeling; researchers have been working on finding
mathematical models that efficiently describe the glioma
growth procedure.
1.1. Mathematical Modelling
A mathematical model uses mathematical language to
describe a system. In the case of glioma, a mathematical
description of how glioma grows is under research. At
first someone has to bear in mind the basic features that
any typical mathematical model in biology must possess
[2]. These are schematically described in Figure 1.
Firstly, the model should be initiated within a realistic
biological state. Additionally, the modeled biological
processes should be understood and discretized as much
as possibly, meaning that steps and real biological para-
meters should be isolated. Continuing, it is essential to
allocate a mechanism that could simulate these steps and
incorporates these parameters. Specifically, this me-
chanism could be described by an equation. Going fur-
ther, the next step is to study the model mathematically
and come up with solutions that include realistic boun-
dary and initial conditions. Lastly, after having acquired
the theoretical results it is of great importance to go back
in biological process with predictions, comments and
suggestions for experiments that will either ascertain or
disprove the developed model. At this level, model suc-
cess is highly dependent on combining experimentation
and theory together. Because, even if the experimental
results indicate that the model is incorrect, this is the
right way to reach a successful conclusion. In final con-
sideration, mathematics is very important in biology,
however, they must be treated with seriousness. If ma-
thematics is used for solving any biological process,
without thoroughly studying the biological background,
it is very possible to come up with solutions that not
only do not contribute to corroborated conclusions, but
also do harm. As stated in [2], the theoretical literature
abounds with many such articles.
Recently, with the explosion of biological sequence
data, many biological sequence databases have redun-

502 A. Roniotis et al. / J. Biomedical Science and Engineering 3 (2010) 501-508
dant sequences which can cause problems for data anal-
ysis. These redundant sequences cannot provide valuable
information for analysis but detracts from the statistical
significance of interesting hits. Moreover, processing
these redundant sequences often requires more time and
computational resources. Removing redundant sequen-
ces is undoubtedly very helpful for performing statistical
analysis and accelerating extensive database searching
[1]. And it is also a way to obtain the real protein fami-
lies and their representatives from a large sequences da-
taset. Therefore, it is necessary to develop an appropriate
algorithm to remove redundant sequences from a bio-
logical sequence database.
Hobohm and Sander’s algorithm is a widely used al-
gorithm in many redundant sequence removing pro-
grams. Hobohm and Sander’s algorithm was firstly in-
troduced by U. Hobohm et al. of EMBL laboratory in
1992. In 1998, Lissa Holm and Chris Sander developed
a program based on this algorithm to generate a non-
redundant protein database NRDB90 [2]. After that, oth-
er researchers developed some programs for removing
redundant sequences on the basis of Hobohm and Sand-
er’s algorithm, such as CD-HIT and PISCES.
A lot of research is currently taking place in mathe-
matically modeling the glioma growth procedure. An
efficient mathematical model for gliomas could help
researchers and clinicians to get a better understanding
of tumour growth pathology. They could also to predict
the aggressiveness of a patient’s tumour and, thus, to
better define the margins of a tumour, so as to use them
Figure 1. Mathematical modeling for a general biological
process.
when applying resection or radiotherapy. Moreover, by
importing therapy parameters into the model, the clini-
cian can predict which therapy scheme is expected to
yield better results for the patient.
1.2. General Cancer Modelling
Studying tumour expansion and simulating this accord-
ing to mathematical models, has been an area of studies
in cancer since late 90s’ [3-5]. Tumour growth has been
studied by a series of models. The first models focused
on tumour behavior in time. More specifically, the first
proposed temporal models were based on either expo-
nential, logistic or Gompertz laws [6]. As expected,
these models were followed by spatial growth models in
later years. Thus, one such deterministic model has been
used to simulate cancer growth as a wave phenomenon,
taking into account mitosis and nutritient depletion [7].
Moreover, deterministic models taking into account im-
mune response [8] or mitotic rates changes [9] have been
proposed.
The research efforts for applying these models in in-
filtrative cancers, such as glioma, failed because cell
motility hadn’t been included in the model. Thus, some
stochastic and cellular automata have been introduced
for glioma simulation, taking into account cell cycle,
lack or abundance of nutritional elements in the sur-
rounding area of cells and therapeutic regimen [10].
However, the mostly used models for the glioma case
are the diffusive models that simulate the spatiotemporal
change of glioma cell density, using partial differential
equations. The combination of biomechanics and diffu-
sive models is one of the latest advances in glioma mod-
eling, for combining diffusion glioma cells and dis-
placement of tissue caused by glioma growth [11].
2. THE DIFFUSION- REACTION MODEL
Unlike solid tumors, for which simple exponential or
geometric expansion represents expansion of tumor vo-
lume, the glioma growth rate cannot be determined as
the classical doubling rate [12], because gliomas can
migrate and proliferate. In order to simulate glioma ex-
pansion, scientists have proposed the application of the
diffusion-reaction equation, which is currently mostly
used. The first that proposed a diffusive model for gli-
oma growth was Murray in 1989 [13]. Murray derived
the equation by exploiting the mass balance equation, by
imagining cells as internal sources for producing the
diffused cells. Murray proposed the diffusion-reaction
formalism as:
(,)
  (,)+(,)((,))















(,)
(1)
where

A. Roniotis et al. / J. Biomedical Science and Engineering 3 (2010) 501-508 503
(,) denotes the glioma cell concentration in posi-
tion  at time .
(,) is the diffusion flux of cell that follows Flick’s
law (i.e. = where  and div are the gradient
and divergence operators respectively).
() denotes the source factor, representing the gli-
oma cell reproduction.
() denotes the treatment factor, representing the
glioma cell loss due to treatment. This is zero, when no
treatment is applied.
The initial state of the model, (, 0), is defined as
the initial distribution of cancerous cells.
() ()() is the net proliferation rate.
This equation was the basis of the later most impor-
tant works for glioma modeling. The general procedure
for glioma modelling, as derived by Murray’s diffusive
model, is presented in Figure 2.
The solution of the diffusion-reaction equation requires
the application of numerical schemes, since there is no
direct formula of its solution. The solution has to be ap-
proximated iteratively till time point of interest is reached.
2.1. Net Proliferation Rates for untreated gliomas
In 1995, Tracqui studied the evolution of cell concen-
Figure 2. Generalised mathematical modeling of glioma
growth.
tration, by using two characteristic of tumour growth:
proliferation and invasion [3]. Tracqui proposed that the
cells proliferate at exponential rate ρ, i.e. :
()= (2)
So, Tracqui changed Equation 1, to

 ()+ (3)
where ρ denotes the proliferation rate of cells. Other
proposed models [14], instead of geometrical rate, used
either Verhulst law ()=
 (4)
or Gompertz law ()=ln 
 (5)
where  is the maximum value that concentration can
reach. Eq.2 has been mainly used for simulating un-
treated gliomas.
2.2. First Estimation of  and 
From the very beginning of diffusive models, one of the
key issues was the estimation of parameters that are be-
ing used. The diffusion coefficient  and the prolifera-
tion rate  are the basic parameters for the diffusive
models. The first estimation of them is places in 1995
[1], when Silbergeld studied biological data and intro-
duced two groups of glioma cells: the common ones and
the resistant-to-fi rst -chemotherapy ones. Parameter D
was firstly defined either at =/, with
the percentage of cells resistant to chemotherapy being
at 8%, or at =/ without resistant cells,
whi le ρ was defined at =/.
2.3. Resection Modeling
Getting this further, one of the next steps was to the
model cancer evolution after ectomy [3,15]. This was
firstly simulated in 1993 by setting the concentration of
the ectomized area equal to zero and, then, allowing the
surrounding malignant cells proliferate and diffuse until
the sphere reaches 6cm diameter. An example of ectomy,
reproduced from [3], is given in Figure 3.
2.4. Low Grade Gliomas
Up to 1996, diffusive models studied high-grade gliomas
due to their remarkably fast invasion. However, studying
low-grade gliomas was important as well. Hence, in
1996, Woodward [16] suggested that speed of growth in
low-grade tumours should be 10% of the respective one
in high-grade gliomas, yielding satisfactory results.
After some years, in 2003, Mandonnet [17] proposed
that low-grade gliomas grew slowly, but linearly. This is
mathematically derivable by Eq.3, because the expand-
ing velocity of a population, which follows only the dif-
fusion and growth laws of (3), can be calculated as

504 A. Roniotis et al. / J. Biomedical Science and Engineering 3 (2010) 501-508
2. Mandonnet et al. used clinical data reproduced
from 27 patients to estimate that the average tumour
velocity was 2 mm per year.
2.5. Brain Heterogeneity
Up to 2000, researchers didn’t take brain anatomy into
account. However, taking into account brain matter is of
foremost importance, since it was observed that migra-
tion in white brain tissue is faster than in grey tissue [4].
Thus, modeling of gliomas should take brain hetero-
geneity into account. Indeed, in 2000, Swanson inno-
vated by introducing the problem and incorporating
white and gray matter differentiation in the diffusion
coefficient  of Eq.3. More specifically, the equation
continued to hold, but for variable , i.e. :

 (())+ (6)
Diffusion coefficient  varies according to position,
with D(x) =  or , i.e. being constant for  in grey
and white brain tissue respectively. Moreover, in order
Swanson to apply the model, an brain atlas was required
providing all the information about white and grey matter
areas. Indeed, BrainWeb database [18] was available for
extracting this information. An example of Swanson’s
simulation, reproduced from [4], is given in Figure 4.
2.6. Anisotropic Cell Migration
In 2005, Jbabdi et al. [19] introduced brain tissue aniso-
tropy in diffusive modeling. As observed, glioma cell
migration is facilitated along the directions of white
matter fibers [20-21]. This observation can be supported
by the diffusion tensor Magnetic Resonance imaging
(DT MRI) that gives a very good 3D reconstruction of
white matter fibers. Jbabdi defined Eq.7 , as

 (())+ (7)
where  is the diffusion tensor that describes cell dif-
fusion rate at point , i.e. a 3 × 3 symmetric positive
definite matrix that reflects local anisotropy.
2.7. Biomechanical Deformation
In 2005, Clatz et al. from INRIA [11] developed a model
that simulates Jbadbi’s model, but taking also into ac-
count the biomechanical deformations that occurs in
brain due to tumour expansion. This model uses a pre-
dictor of the mass effect induced by both the tumor pro-
liferation and infiltration. Figure 5 shows an example of
a simulation with Clatz’s model, with the deformation of
tissues presented. The image is reproduced from [11].
Figure 3. (Reproduced from [3])-tracqui’s simulation of a tumour resection. The parameter values of the model are  =
0.012 /day,  = 10-7 cm2/s.
(a) (b) (c) (d)
Figure 4. (Reproduced from [4])- Simulation of tumour invasion of a high-grade glioma in the superior cerebral hemisphere
using Swanson’s model: (a) (b) at diagnosis; (c) (d) at death; (a) (c) as seen by Swanson’s standard threshold of detection; (b)
(d) as calculated out to 1.25% of the threshold (boundary) cell concentration defining the sensitive threshold of detection.

A. Roniotis et al. / J. Biomedical Science and Engineering 3 (2010) 501-508 505
Figure 5. (Reproduced from [11])-Simulation of displacement of the tissues induced by the tumor mass effect, us-
ing Clatz’s model.
3. Parameter estimation in modern models
According to [22], Figure 6 can be used as a guiding
index for defining parameters  and  according to
glioma grade, velocities of growth and /. Indeed,
this log to log graph includes all parameter values found
up to Jan. 2007 for both low-and high-grade gliomas.
Low-grade gliomas are sited in bottom left rectangle
(LGG), for 2 mm/yr average velocities. Respectively,
high grade gliomas (HCG) are positioned in the large
rectangle, defined by / of 2 to 20 cm2 and average
velocities from 10mm/yr to 200mm/yr. On the left part
gliomas with detectable mass, which can be cured with
surgery, are placed.
Figure 6. (Reproduced from [22])-Log-log graph of  and ,
for high-and low-grade gliomas.
Continuing, for the case of heterogenous brain matter
for high-grade gliomas, it is suggested in [4] that a typi-
cal value for  is =0.0012 /day. This value for low-
grade gliomas can be defined at  = 0.00012 /day. Then
by assuming that =5, =102 mm2/day and
=2 103 mm2/day can be used [19].
It is noteworthy to present the following table, repro-
duced from [23], where there is a list of the proposed val-
ues of each parameter that the latest glioma models use.
4. CHEMOTHERAPY MODELING
Chemotherapy modeling is a quite blurred part of glioma
models, which is mostly studied currently by researchers.
Parameters that have to be taken into consideration
Table 1. (Reproduced from [23])-The parameters of the diffu-
sive model as proposed in the last 2 decades.
Parameter (namely) Parameter
Symbol Value Reference
Growth Rate

0.012
(1/day)
[24]
Diffusion Coefficient
(Gray matter)

0.0013 (
cm2/day)
[3]
Diffusion Coefficient
(White matter)

5

(
cm2/day)
[4]
Initial number of
tumor cells
0
105 cells
[16,25]
CT threshold density - 400 (cells/
mm2)
[4]
CT threshold radius - 1.5
cm
[24]
Cell death rate
(chemotherapy)

0.0196
(1/day)
[26]
Number of
fractions/ day

1-conventional (CR)
2-hyperfractionated
(HF)
[27,28]
Time interval (step)

1
day
(CR)
6
hours
(HFR) [ 29,30]

506 A. Roniotis et al. / J. Biomedical Science and Engineering 3 (2010) 501-508
should be extracted by histopathological and biostatis-
tical data of specific patients [31]. Swanson has intro-
duced a generalised net proliferation rate of chemothe-
rapy as:
()= () (8)
where () is the temporal profile of the chemotherapy
treatments, assuming a loss proportional to the strength
or amount of therapy at a given time. Swanson sets
()= when the chemotherapy is being administered
and ()= 0 otherwise.  is actually a measure of
effectiveness of chemotherapy. Thus, in order therapy to
be effective and size of tumor to decrease,  should be
larger than , so that the net proliferation rate () is
negative. This means the number of the dying cells is
larger than the new born cells.
An example of chemotherapy application on real clin-
ical MRI data, using (8) can be reproduced by [32]. In
this example, k was intentionally set to the high value of
()=0.024, so as the cancer to have shrinking effect.
The data (18 MRI slices) has been acquired by Univer-
sität des Saarlandes Klinikum (Germany) within the
scope of the Contracancrum project [33]. The modeling
results are given in Figure 7. The 3-dimensional repre-
sentation of the initial and after-chemotherapy states are
presented, accompanied by a sampled series of 4 MRI
slices.
4.1. Mathematical Frameworks
The increasing interest of researchers on diffusive mod-
els was remitted by the total absence on specific guide-
lines on how to design them mathematically. In 2009,
Roniotis et al. [34] published a mathematical framework
on designing diffusive models by using Finite Differ-
ences, for the needs of the ContraCancrum Project [33].
The framework includes heterogeneous tissue, aniso-
tropic migration of cells along white fibers and is 3D.
Moreover, chemotherapy formalism of Eq.8 can also be
incorporated in the model.
Different schemes of finite differences have been de-
veloped, namely forward Euler, backward Euler, Crank
Nikolson and θ-methods. Moreover, the same model has
been also developed with finite elements. The accuracy
of the different schemes has been tested on simplified
cases which suggested that the backward Euler scheme
(Finite Differences) yields the best results. An example
of applying this framework on real clinical data is dis-
played in Figure 8, reproduced from [34].
5. CHALLENGES FOR FUTURE MODELS
Simulations of applying therapy can be performed by
adjusting the proliferation term according to glioma
cell proliferation rates in the diffusion equation. More-
over, consecutive chemotherapy sessions have been
simulated by works. However, the main issue govern-
ing these applications is the estimation of the most
efficient model parameters. This requires study of
pharmacokinetics and study of real chemotherapy ses-
sions on patients.
(a)
(b)
Figure 7. (Reproduced from [32])-MRI and 3D representation of glioma in. (a) initial state; (b) after chemotherapy
simulation. 4 slices out of 18 slices are presented.

A. Roniotis et al. / J. Biomedical Science and Engineering 3 (2010) 501-508 507
Figure 8. (Reproduced from [34])-results of modeling (left column) initial data, (central column) simulated data after 112
days, by using the backward euler scheme and (right column) real data on 112th day.
An ideal model will have the ability to suggest paths
of invasion, unseen by the doctor and help the clinician
predict how the tumor is going to behave after different
cures. Personalized parameter estimation for treatment
term, proliferation rate and initial state are very impor-
tant for the model and are being studied by most re-
searchers working on diffusive models globally.
Another important issue regarding modern diffusive
models is the estimation of parameters of anisotropic
diffusion. This is already incorporated in the models, but
the methods used for extracting diffusion tensors have
not been validated. This is mostly due to the use of DT-
MRI, that is a modern technique and is hard to accom-
pany all medical data. An estimation of tensors based on
atlas could be studied.
6. CONCLUSIONS
Mathematical diffusive modeling for simulating glioma
growth and invasion is a currently developing scientific
field. An efficient 3-dimensional model of glioma
growth will constitute a powerful tool for clinicians,
since they could predict how glioma is going to develop
in time. The most important part of this is the differen-
tiation of the predicted outcome according to the differ-
ent model input parameters. These parameters are
changing with different therapy treatments. Thus the
clinician can make an easier decision on which therapy
scheme yields the best predictions.
7. ACKNOWLEDGEMENTS
This work was supported in part by the EC ICT project ContraCan-
crum, Contract No: 223979.
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