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Abstract— Glioma is the most aggressive type of brain
cancer. Several mathematical models have been developed
towards identifying the mechanism of tumor growth. The
most successful models have used variations of the
diffusion-reaction equation, with the recent ones taking into
account brain tissue heterogeneity and anisotropy.
However, to the best of our knowledge, there hasn’t been
any work studying in detail the mathematical solution and
implementation of the 3D diffusion model, addressing
related heterogeneity and anisotropy issues. To this end,
this paper introduces a complete mathematical framework
on how to derive the solution of the equation using different
numerical approximation of finite differences. It indicates
how different proliferation rate schemes can be
incorporated in this solution and presents a comparative
study of different numerical approaches.
I. INTRODUCTION
LIOMAS are the most malignant form of brain
tumor, which differ from other tumors, because of
their highly aggressive and diffusive behavior. Since
90s, many important diffusive models have been
introduced [1-3], with the most recent ones taking brain
tissue heterogeneity and anisotropic cell migration into
account. According to Jbadbi [3], the spatiotemporal
diffusion equation that describes glioma growth is
(1)
where c(x,t) is the tumor concentration in position x at
time t, is the diffusion tensor, i.e. a 3x3 symmetric
matrix that expresses anisotropy of cell migration, and
div are the gradient and divergence operators
respectively and is the net cell proliferation rate.
Variant formalisms of have been proposed [4], with
the main ones following either the exponential law
, (2)
or the Verhulst law
(3)
or Gompertz law
(4)
Manuscript received April 7, 2009. This work was supported in part by
the EC ICT project ContraCancrum, Contract No: 223979.
A. Roniotis, V. Sakkalis, and K. Marias are with the Institute of
Computer Science, Foundation for Research and Technology
(FORTH), Heraklion 71110, Greece (e-mail: {roniotis; sakkalis;
marias}@ics.forth.gr).
G. D. Tsibidis is with the Institute of Electronic Structure and Laser,
Foundation for Research and Technology (FORTH), Heraklion 71110,
Greece (e-mail: tsibidis@iesl.forth.gr).
A. Roniotis and M. Zervakis are with the Dept. of Electronic &
Computer Engineering, Technical University of Crete, 73100, Chania,
Greece (email: roniotis@ics.forth.gr, michalis@display.tuc.gr).
where denotes the proliferation rate constant and is
the maximum value that can reach.
Due to the spatial dependence of D, an analytical
solution of (1) cannot be acquired; therefore, it has to be
numerically approximated. Finite differences (FDs) are
commonly used in diffusive models. By using FDs, a big
system of equations arises, the iterative solution of
which yields the approximated tumor cell concentration
at a desired time point. Different numerical schemes for
approximating partial derivatives are able to differentiate
the emerging system, thus the way it is solved.
Up to now, the various implementations lack a firm
mathematical background on the derivation of the
system, with concrete assumptions on the approximation
scheme. The main objective of this paper is to provide
the direct formalism of the derived linear system, for the
widely used FD schemes, namely forward Euler (FE),
backward Euler (BE), Crank Nikolson (CN) and θ-
methods. These formalisms are designed for 3D,
heterogeneous and anisotropic brain tissue and entail the
general form of , so that one could use any net
proliferation rate. We also present how equations (2),(3)
and (4) are tailored to the model. Finally, we make a
comparative study of the different FD numerical
schemes by analyzing experimental results.
II. METHODS
Equation (1) is a partial 2nd order differential equation.
Before hunting up a direct expression of the linear
system that iteratively solves it, (1) is expanded in three
spatial dimensions. Assume that the diffusion tensor
D(x) at a point x=(x1, x2, x3) is:
(5)
Then, by using definitions of and , (1) becomes:
(6)
The next step is to use FDs so as to approximate the
solution of (6). For approximating the partial derivatives,
i) FE, ii) BE, iii) θ- and iv) CN schemes are used.
A. Forward Euler Method (FE)
At first, assuming that initial and boundary conditions
have been defined, we study how FE approximates the
solution of (6) for a grid. If
c(nΔT,iΔX,jΔY,kΔZ)≡, Dpq(iΔX,jΔY,kΔZ)≡,
i , j , k and
, then the partial derivatives of (6) in point
A complete mathematical study of a 3D model of
heterogeneous and anisotropic glioma evolution
Alexandros Roniotis, Kostas Marias, Vangelis Sakkalis, George D. Tsibidis, and Michalis Zervakis
G
(iΔX,jΔY,kΔZ) can be approximated as
(7)
and similarly for , , , and .
If any of the neighboring is a boundary point, then
its value will be as initially defined. Continuing, by
substituting the approximations (7) to (6), we derive:
+ +
+ + +
+ + +
+ + +
+ + +
+ + +
+ + (8)
or equivalently:
(9)
where
(10)
If the vectorized version of at time m is taken, as
(11)
then, the overall solution of the equation at time ,
for given , can be found by solving iteratively
where A is a
atrix with its elements defined as:
where:
(12)
and F a vectorization operator that vectorizes f( ) as
(13)
A is a symmetric, sparse, 19-
diagonal matrix, with its form being visualized in Fig.1,
where all light gray areas have zero values.
After having acquired A, a direct solution can be found
by iteratively calculating
(14)
where I is the identity matrix. This is
the solution of the FE. This is called forward, because
the next-time approximation of concentration can be
directly estimated as a linear combination of the
previous-time approximation and is easy to implement,
but numerical stability has to be ensured. As proven in
[5], this method is stable giving reliable results when
(15)
B. Backward Euler (BE)
Similarly, the density of glioma cells at time t, using BE,
can be derived by iteratively solving the system:
- (16)
till time is reached, with A and F being the same with
(12) and (13) respectively. In each iteration the system
can be expressed with a big, sparse, symmetric and
positive definite matrix; an iterative method for solving
Fig. 1. Sparse Matrix A: White areas have zero values, thick lines
are 3 diagonals in a row and thin lines are single diagonals.
linear systems, e.g. the conjugate gradient method can be
used. Unconditional stability and accuracy are
advantages of this method, whereas computational and
storage load that has to be considered [6].
C. θ-methods/Crank Nikolson (CN)
The θ-methods use a balancing parameter θ
so as to combine forward and backward numerical
schemes concurrently. By using θ-methods we get
- (17)
where . As in BE, a method for solving linear
systems is required. The θ-methods are flexible due to
choice of parameter θ but higher computational and
storage load are required [6]. CN is a θ-method for θ=½.
In Table I, the formulation of the iterative linear
systems that produce the approximated solution of (6) is
presented according to various selected methods. Based
on this, we have implemented models for glioma growth,
directly in 3D using C++ (the authors can be contacted
for more details).
D. Vectorization Operator
It is interesting to study how operator F is chosen for
the mainly used proliferation rates of (2),(3) and (4). If
the net proliferation rate is given by (2) holds, then
according to (13):
(18)
or equivalently:
(19)
Next, if the net proliferation rate is given by (3), then:
(20)
or equivalently:
. (21)
The definitions of and are given in Table 2.
Lastly, if the net proliferation rate is given by (4) holds,
then according to (13):
(22)
or equivalently:
(23)
In Table II, a summary of vectorization operator for the
commonly used proliferation rates is presented.
III. RESULTS AND DISCUSSION
In order to study the performance of the different
numerical schemes presented, a simplified test case of
the pure diffusion equation is used, for which there is a
known analytical continuous expression of the solution.
Hence, the magnitude of each numerical scheme
deviation from the real solution can be studied, which
serves here as ‘ground truth’ for validating our numerical
approximations.
A. Spherical homogeneous test tumor
To validate our methodology, it is assumed that tumor
growth in 3D, unbounded, isotropic (i.e. D is constant)
and homogeneous region exhibits a pure diffusion
behavior (i.e. ). The tumor has initially a
concentration and it is constrained in a sphere of
radius (Fig. 2). Due to symmetry, the concentration of
glioma depends only on the distance from the center of
the sphere and it is given by the expression [7]:
(24)
B. Testing and Simulation Description
In our tests, the model is adapted to approximate (24),
by using the following parameters:
, , ,
, points and
. Five different simulations were
run for 200 days, using FE, three θ-methods for θ=0.25,
θ=0.5 (CN) , θ=0.75, and BE performed on a Pentium 4
at 3.8 GHz.
In Fig.3, the evolution of cell concentration in time is
presented, with respect to , with lines corresponding to
results from analytical expression (24), while dots
representing what simulation with CN yields. Fig. 3
shows a very good agreement between these results;
however a more rigorous investigation would require an
error estimation analysis.
Fig. 2. Left: The initial spherical tumor of radius . Right: The
initial tumor concentration according to at time t=0
TABLE I
DIRECT EXPRESSIONS OF THE ITERATIVE LINEAR SYSTEMS
Numerical
scheme
Iterative linear system solving (6)
Forward
Euler
Backward
Euler
θ-methods
TABLE II
VECTORIZATION OPERATOR FOR DIFFERENT PROLIFERATION RATES
Proliferation
Rate
Constan
t Rate
f(c)
Vectorization Operator
Exponential
Verhulst
(logistic)
Gompertz
C. Error Estimation
In order to estimate the error that each scheme yields
the normalized mean absolute error is introduced. Due
to symmetry, at time is computed as
100% (25)
In Table III, the estimated for each scheme on day 10,
100 and 200 of simulation is reported. Moreover, the
overall simulation times are also reported. The highest
for all five schemes is on 10th day, while it decreases to
values on the 200th day. FE error is the lowest
on 10th day, i.e. 6.36%, but overcomes all other schemes
on 200th day, reaching 0.57% at the 200th day. CN is the
scheme that reaches the lowest error value at 200th day at
0.23%. In Fig.4, a logarithmic graph of in time is
presented, for each scheme. It is noticeable that all
schemes initially produce higher errors. A possible
explanation is that the initial concentration has
an abrupt step descent at r=10mm, as seen in Fig. 2
(right). This makes the approximation of the partial
derivatives at the edge of this step erroneous for all
schemes. However, as simulation continues, all schemes
tend to significantly decrease . Continuing, one can
observe that decreases to values <1% on day 40, 75,
94, 143 and 156 for FE, 0.25-method, CN, 0.75-method
and BE respectively. FE error has a remarkable descent
till day 42, but later decreases smoothly and finally
overcomes all other schemes. BE produces the highest
error, till day 186, when FE overcomes it. As expected,
approximation with θ-methods is more accurate than BE
throughout all days and more accurate than FE in most
time. θ-method for θ=0.25 initially tends to have a steep
descent to error 0.3%, but later reaches a balancing value
higher than CN. θ-method for θ=0.75 has a smoother
descent, being always higher than CN. Lastly, CN
(θ=0.5) seems to have the best general performance,
reaching the lowest error 0.23% on day 200.
Thus, CN seems to yield more accurate approximations
of . However, if there is room for sacrificing
accuracy for faster model simulations, then FE should be
used, since from Table III it’s derived that FE is 4.63
times faster than CN. Even if one chooses FE, that is the
worst case scenario, error doesn’t overcome 0.6%
IV. CONCLUSION
This paper presents a complete, FDs’ based,
framework for the mathematical solving of the
anisotropic, heterogeneous and 3D diffusion-reaction
equation simulating either the growth of glioma or other
phenomena where diffusive models are applicable.
Moreover, the introduction of the vectorization operator
gives the flexibility to include any proliferation rate into
the model. Lastly, a performance study of different
numerical schemes is presented based on an analytical
expression that can be derived for the pure diffusion
equation.
REFERENCES
[1] P. Tracqui, “From passive diffusion to active cellular migration in
mathematical models of tumor invasion,” Acta Bibliotheoretica,
vol. 43, pp 443-464, 1995.
[2] K. R. Swanson, E. C. Alvord, and J. D. Murray, “A Quantitative
Model for Differential Motility of Gliomas in Grey and White
Matter,” Cell Proliferation, vol. 33, no. 5, pp. 317-330, 2000.
[3] S. Jbabdi, E. Mandonnet, and H. Duffau, “Simulation of
anisotropic growth of low-grade gliomas using diffusion tensor
imaging,” Magn Reson Med, vol. 54, pp. 616-24, 2005.
[4] M. Marusic, Z. Bajzer, J. P. Freyer, and S. Vuk-Palovic,
“Analysis of growth of multicellular tumour spheroids by
mathematical models,” Cell Prolif, vol. 27, pp. 73-94, 1994.
[5] S. Puwal, and B. J. Roth, “Forward Euler Stability of the
Bidomain Model of Cardiac Tissue,” IEEE Transactions on
Biom. Eng., vol. 54, no. 5, pp. 951-953, 2005.
[6] S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes:
The Art of Scientific Computing (Third Edition), Cambridge
University Press, NY, 2007.
[7] J. Crank, The Mathematics of Diffusion (Second Edition), Oxford
University Press, NY, 1975.
TABLE III
ERROR AFTER 10, 100 AND 200 DAYS
Scheme
10 days
100 days
200 days
Simulation Time
FE
6.36%
0.89%
0.57%
3’53”
θ-method
(θ=0.25)
9.73%
0.43%
0.33%
18’13”
CN
9.83%
0.91%
0.23%
19’01”
θ-method
(θ=0.75)
10.17%
1.43%
0.35%
20’16”
BE
11.33%
1.95%
0.50%
22’31”
Fig. 3. The evolution of after 10, 100 and 200 days
70 80 90 100 110
101
102
103
104
r(mm)
C(r)
cells/mm 3
Analytical Solution
Numerical Solution
Analytical Solution
Numerical Solution
Analytical Solution
Numerical Solution
Day 200
Day 100
Day 10
Fig. 4. Logarithmic visualization of error in time
