An ADER finite volume method for an
atherosclerosis model
Arturo Hidalgo and Lourdes Tello
Universidad Politécnica de Madrid (Spain).
International Workshop on Recent Advances in Numerical
Methods for Hyperbolic Conservation Laws and Nonlinear
Time Dependent Partial Differential Equations in Honour
of
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
6. Characterization of initial conditions in
the phase plane.
7. 2D Case.
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
Source:
Is an inflammatory disease caused by the deposition of low
density lipoproteins (cholesterol LDL) in the walls of the
arteries.
WHAT IS ATHEROSCLEROSIS ??
FIRST STAGES IN ATHEROSCLEROSIS
FIRST STAGES IN ATHEROSCLEROSIS
FIRST STAGES IN ATHEROSCLEROSIS
FIRST STAGES IN ATHEROSCLEROSIS
FIRST STAGES IN ATHEROSCLEROSIS
FIRST STAGES IN ATHEROSCLEROSIS
FIRST STAGES IN ATHEROSCLEROSIS
Foam cells set up a chronic inflammation by
secreting pro-inflammatory cytokines (TNF-
,IL-1,…).
FIRST STAGES IN ATHEROSCLEROSIS
Monocytes.
(Source: Wikipedia)
Macrophages and foam cells formation.
(Source:
Foam cells in aorta. (Source:
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
6. Characterization of initial conditions in
the phase plane.
7. 2D Case.
Hidalgo, A., Tello, L., Toro, E.F.
Numerical and analytical study of an
atherosclerosis inflammatory disease model
. Journal of Mathematical
Biology. June 2014, Volume 68, Issue 7, pp 1785–1814.
where El Khatib’s model was followed…
Starting point
Philos Trans A Math Phys Eng Sci.
2009 Dec 13;367(1908):4877-86. doi:
10.1098/rsta.2009.0142.
Mathematical modelling of atherosclerosis as an inflammatory disease.
2 2 1 2 2 2 2 0 1 0 2
1
( )
,
(0, ) ,
0
( )
,
(0, ) ,
0
0,
0 on
0 and
( , 0)
( ),
( , 0)
( ).
x
L t
t
x
x
L t
A
A
A
t
x
x
M
M
M
d
A
A
A
A
f
x
L
x
x
x
x
x
A
x
d
M
M
M
M
f
MATHEMATICAL MODEL
M
(x,t)
density of immune cells (monocytes, macrophages);
A
(x,t)
density of cytokines secreted by immune cells;
t
time;
x
length along wall of the artery.
d
1,
d
2
diffusivity of the immune cells and cytokines, respectively.
1,
2
rate of degradation of the immune cells and the cytokines,
respectively
1 1
1 1
( )
1
A
A
A
f
MATHEMATICAL MODEL
is the recruitment of the immune cells from the blood flow.
2 2
2
( )
1
f
A
A
A
REMARK
In the reference: Montecinos G.I. and Toro E.F.
Reformulations for general
advection–diffusion–reaction equations and locally implicit ADER
schemes
. JCP,2014;
following
Cattaneo's original idea
the authors present two relaxation
formulations for time-dependent,
non-linear systems of advection–
diffusion–reaction equations
. Such formulations yield time-dependent
non-linear hyperbolic balance laws
with
stiff source terms
.
MATHEMATICAL MODEL
Now we considered a modified model in which diffusion is nonlinear
Porous medium-type.
2 2 1 2 2 2 2 0 0 1 1 2( )
,
(0, ) ,
0
( )
,
(0, ) ,
0
0,
0 on
0 and
( , 0)
( ),
( , 0)
( ).
m mM
x
L t
t
x
x
L t
t
x
x
M
M
M
M
A
A
A
d
A
A
f
f
x
L
x
x
x
x
x
x
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
6. Characterization of initial conditions in
the phase plane.
7. 2D Case.
1
( )
( ) :
( )
Q
Q
D
Q
R Q
Q
m
Q
mwi h
x
t
t
x
1 21 1 2 2
0
(
, ) ,
0
( ( )
,
( )
)
Q
D
R
T Td
M A
d
f A
M f A M
A
where
1
( )
( ) :
( )
Q
Q
D
Q
R Q
Q
m
Q
mwi h
x
t
t
x
1 21 1 2 2
0
(
, ) ,
0
( ( )
,
( )
)
Q
D
R
T Td
M A
d
f A
M f A M
A
where
Let us consider the space-time control volume
1 1/ 2 1/ 2
[
i,
i] [ ,
n n]
V
x
x
t t
1
1/ 2 1/ 2
n n
i i i i i
t
t
x
Q
Q
G
G
R
and integrate over the control volume V
1
1/ 2 1/ 2
n n
i i i i i
t
t
x
Q
Q
G
G
R
1 1/2 1/2 1 1/2 1/2 1
1/2 1/2 1/2
1
1
( , )
,
(
, )
(
, )
1
( ( , ))
Q
Q
Q
G
D
Q
R
R Q
n i n i n i n i x t n n
i i i i
x t
m
t x
i t x
x t dx
x
t
x
t dt
x
t
x
x
m
t dxdt
x t
where
NUMERICAL APPROXIMATION: FV framework
The intercell flux at x
i+1/2
is given by an integral average in
time of the solution
1/2 1
1/2 0 /2
1
(
(
, )
)
(
, )
Q
G
iD
tQ
i id
t
x
x
x
Therefore we need the values
NUMERICAL APPROXIMATION: Main ingredients
•ADER schemes were first introduced in Toro et al. 2001 for hyperbolic problems.
•Applied to reaction-diffusion:
E.F Toro and A. Hidalgo
,
"ADER finite volume schemes for nonlinear
reaction-diffusion equations "
APNUM, 2009.
G. Gassner, C.-D. Munz, F. Lörcher,
" A contribution to the construction of diffusion
fluxes for finite volume and discontinuous Galerkin schemes “
JCP, 2007.
A. Hidalgo and M. Dumbser
,
“ADER Schemes for Nonlinear Systems of Stiff
1
1/2 0
1
(0, )
Q
G
iD
tm
Q
md
t
x
We wish to compute the intercell numerical flux:
with high accuracy in time. The procedure to achieve it
contains two main ingredients:
(i)a high-order nonlinear spatial reconstruction of the
gradient of the solution in each cell and
(ii)the solution of the generalized (or high-order)
Riemann problem at the interface of each cell.
NUMERICAL APPROXIMATION: High-order nonlinear reconstruction
We use Weighted Essentially Non Oscillatory (WENO)
reconstruction (
see, e.g., Balsara and Shu JCP 2000,Titarev
and Toro, JCP 2005
)
We need to obtain high order polynomials which allow us to
achieve values and derivatives where needed.
In WENO reconstruction for an order of accuracy
r
we
have
r
candidate stencils each one with
r
cells
6 k kk r 1 k p
k j
j 0
d
with
;
(k
0,1,...,r 1),(
10 ;
p
2)
i 1/ 2
i 1/ 2
2
x m
r 1
2m 1
k m k
m 0 x
d
p
x
x
dx
(k
0,...,r 1)
dx
with smoothness indicators
The linear weights, d
i
, used in this work are based on
Dumbser, Enaux, Toro (2008) where a very large weight
is assigned to the central stencil and a very small linear
weight is assigned to the biased ones.
The reconstruction polynomial for cell i is then obtained as
a convex combination of the
r
polynomials
P
l
taken with
positive nonlinear weights. These weights are
NUMERICAL APPROXIMATION: FV framework
The intercell flux at x
i+1/2
is given by an integral average in
time of the solution
1/2 1
1/2 0 /2
1
(
(
, )
)
(
, )
Q
G
iD
tQ
i id
t
x
x
x
Therefore we need the values
(0)
( )
Q
LR
ii 1(x,t)
( (x,t))
(x,t)
( (x,t)),
x
,t
0 ,
t
x
x
(x)
if
x
0,
(x,0)
(x)
if
x
0.
Q
D
Q
Q
R Q
P
Q
P
where
P
i
and
P
i+1
are polynomials to be obtained via a
reconstruction procedure (WENO in this case).
The sought solution is obtained using the following power
series expansion
k k(0) (
L K K 1 x k k 1 R 0)
(0,0 )
(
)
k! t
0 )
( )
(0,
Q
Q
Q
The leading term is obtained solving the linearized problem
2 2 i i 1ˆ
(x,t)
(x,t),
x
,t
0 ,
t
x
(0)
if
x
0,
(x,0)
(0)
if
x
0.
Q
D
Q
P
Q
P
NUMERICAL APPROXIMATION: ADER approach
(0)
( )
NUMERICAL APPROXIMATION: ADER approach
The solution reads
(0)
i i 1
1
( (0)
(0))
2
(0,0 )
P
P
Q
The high order terms are obtained using the
Cauchy-Kowalewskaya procedure.
(0)
( )
k(0) (k ) (0) (2) (2k )
x x x
k
(0,0 )
H
(0,0 ),
(0,0 ),
,
(0,0 ) .
t
Q
Q
Q
Q
And solve the following classical Riemann problem
Being the solution
m
m
m1 d
d
(0,0 )
(
(0)
(0)).
Q
P
P
j 2 j
j 2 j
j k
i
k j
i 1
(x,t)
(x,t) ,
x
,t
0 ,
t
x
x
x
(0)
if
x
0 ,
(x,0)
x
(0)
if
x
0 .
Q
D
Q
P
Q
P
NUMERICAL APPROXIMATION: ADER approach
(0)
( )
(1)
( )
Q
LR
2 2 i i 1(x,t)
( (x,t))
(x,t)
( (x,t)),
x
,t
0 ,
t
x
x
x
d
(x)
if
x
0,
dx
(x,0)
x
d
(x)
if
x
0.
dx
Q
D
Q
Q
R Q
P
Q
P
where
P
i
and
P
i+1
are polynomials to be obtained via a
reconstruction procedure (WENO in this case).
(1)
( )
Q
LRThe sought solution is obtained using the following power
series expansion
k k(1 (1) K K 1 k L )
R k x
1
(0
(0,0
( )
,0 )
(
)
k! t
)
Q
Q
Q
The leading term is obtained solving the linearized problem
2 2 i i 1ˆ
(x,t)
(x,t) ,
x
,t
0 ,
t
x
x
d
(0)
if
x
0,
dx
(x,0)
x
d
(0)
if
x
0.
dx
Q
D
Q
P
Q
P
NUMERICAL APPROXIMATION: ADER approach
The solution reads
i
i 1(0)
1 d
d
(
(0)
(0))
2 dx
dx
(0,0 )
P
P
Q
(1)
( )
Q
LRNUMERICAL APPROXIMATION: ADER approach
k(1) (k ) (0) (2) (2k )
x x x
k
(0,0 )
P
(0,0 ),
(0,0 ),
,
(0,0 ) .
t
Q
Q
Q
Q
And solve the following classical Riemann problems
Being the solution
j
j
j1 d
d
(0,0 )
(
(0)
(0)).
Q
P
P
j 2 j
j 2 j
j
k j i
k j
i 1 j
(x,t)
(x,t) ,
x
,t
0 ,
t
x
x
x
d
(0)
if
x
0 ,
dx
(x,0)
x
d
(0)
if
x
0 .
dx
Q
D
Q
P
Q
We also need to compute the integral of the source term
i 1/2 i 1/2N M
t x
r s r s
0 x
r 1 s 1
( (x,t))dxdt
( (x ,t ))
R Q
R Q
In order to obtain
Q
(x
r
,t
s
) we apply the following
Taylor expansion
K k k K 1r r k r
k 1
(x , )
(x ,0)
(x ,0)
(
) ,(r
1,
,N;s
1 ,M)
k! t
Q
Q
Q
and we use again Cauchy-Kowalewskaya procedure to express time
derivatives as functions of space derivatives.
The leading term and high order terms are then calculated by means
of spatial derivatives of the reconstruction polynomial inside the cell
i.
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
2 2 x 4q(x,t)
q(x,t)
1 q(x,t)
0.5q(x,t)
S(x,t), x
( 10,10),t
0
t
x
x
q
0, on x
10 and x
10
x
x
q(x,0)
e
sin
20
In order to carry out a convergence test analysis we use the following auxiliary
nonlinear reaction-diffusion problem:
2 2 2
2 2 2
2
x x x
t t t
4 4 4
2
x x 2 x 2
t t t
4 4 4
x
x
x
S(x,t)
e
2 e
sin
e
cos
20
2
20
20
x
1
x
x
1
e
sin
e
e
sin
20
2
4
400
20
2 x t 41
x
e
sin
4
20
where the forcing term is:
And the analytical solution is given by
Comparison of the analytical solution (full) and the numerical solution (symbols) for
Convergence rates for auxiliary test problem
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
6. Characterization of initial conditions in
the phase plane.
7. 2D Case.
Paremeters for the numerical simulation
x=0.01,
t=min(
(
x)
2/d
1
,
(
x)
2/d
2) with
=0.45
Bistable case: Two steady-state state solutions depending on the size
of the initial condition.
Monostable case: One steady-state state solution .
1
2
1
1
2
1
2d
1d
2Density of cytokines (
A
(x,t))
Numerical results: Bistable case. Small perturbation of healthy
steady state. Case
m=2
x
t
x
t
Numerical results: Bistable case. Large perturbation of healthy
steady state. Case
m=2
Density of cytokines (
A
(x,t))
x
t
Numerical results: Bistable case. Small perturbation of healthy
steady state. Case
m=3
Density of cytokines (
A
(x,t))
x
t
x
t
Numerical results: Bistable case. Large perturbation of healthy
steady state. Case
m=3
Density of cytokines (
A
(x,t))
x
t
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
6. Characterization of initial conditions in
the phase plane.
7. 2D Case.
Steady solution (7.14, 6)
This task is computer-intensive since it implies the computation
of the steady state solution using the iterative procedure for
SOME ASPECTS OF THE ANALYTICAL STUDY
We have proved some results regarding some properties of solutions of (P) and their
evolution.
1 1 2 2,
1
1 1 1
0,
1 1 2 2 1 1 1 2 1
,
CONTENTS
1. Biological motivation.
2. Mathematical model.
3. Numerical approximation.
4. Convergence test for auxiliary reaction-
diffusion problem.
5. Numerical examples.
6. Characterization of initial conditions in
the phase plane.
7. 2D Case.
2D MATHEMATICAL MODEL
0
L
ɛ
0
y
A
y
M
0
0
x
A
x
M
0
0
x
A
x
M
1 1( )
;
0
M
d
A
A
f
y
y
Intima
Blood flow
1 1 1 1( )
1
A
A
A
f
1 2 1 2 2 1
1
