Coupled vs decoupled approaches for PDE/ODEsystems modeling intercellular signaling

Thomas Carraro1, Elfriede Friedmann, Daniel Gerecht

Institute for Applied Mathematics, Heidelberg University, INF 293/294, 69120 Heidelberg,Germany

Abstract

We consider PDE/ODE systems for the simulation of intercellular signaling in

multicellular environments. Since the intracellular processes are described by

few ODEs per cell, the PDE part dominates the solving effort. Thus, it is not

clear if commonly used splitting methods can outperform a coupled approach.

Based on a sensitivity analysis, we present a systematic comparison between

coupled and decoupled approaches for this class of problems and show numerical

results.

Keywords: Coupled PDE/ODE systems, Sensitivity analysis, Multilevel

preconditioner, Intercellular signaling

1. Introduction

Cellular signaling has been mathematically described by a variety of different

models mostly relying on large systems of ordinary differential equations (ODE)

[1]. These earlier models were extended by partial differential equations (PDE)

to accurately consider the spatial concentration distribution, localized effects5

and signal ranges [2, 3, 4, 5]. In these cellular signalling models the PDE part

captures the diffusion of the signaling proteins and the ODE part the chemical

processes in some parts of the domain which can be considered in a specific

context as well mixed (e.g. cell surface, nucleus). The coupling between these

1Corresponding authorE-mail address: thomas.carraro@iwr.uni-heidelberg.de

Preprint submitted to Journal of Computational Physics April 17, 2015

parts occurs in linear and nonlinear terms and through Robin-type boundary10

conditions. The analysis and numerics of such models can not be investigated

by standard methods and depends strongly on the structure of the equations

[6]. In this paper we focus on the numerical treatment of such coupled systems.

There are mainly two strategies for implicit solvers of coupled systems: (1)

nonlinear methods among them the nonlinear multigrid method also called15

“full approximation scheme” (FAS) [7, 8], (2) linearization based approaches

(Newton-type). These methods can be used in a combined approach, where for

example a Newton-type method is applied as smoother for a FAS or a linear

(or nonlinear) multigrid method is used as a preconditioner for a Newton-type

method. Our intention in this work is not particularly the comparison and20

discussion of advantages and disadvantages of these strategies that depend on

many aspects like, e.g. the accuracy of the Jacobian approximation [9]. More-

over, since in the considered coupled PDE/ODE system the linearization is not

a critical point we choose a Newton-type method preconditioned by a linear

multigrid method and study the effect of splitting the PDE part from the ODE25

part in the linearized system of equations. A splitting solution approach is

often used when restrictions on accuracy can be relaxed in order to allow an

easier numerical treatment of complicated problems. Such an approach makes

it possible to reuse existing solvers and is widely used in numerical methods

for coupled systems, see [10, 11, 12, 13, 14, 15]. In case of strongly coupled30

equations, this strategy can only be implemented at high computational costs

through very small time steps or a higher number of iterations in the splitting

scheme. We consider PDE/ODE systems for the simulation of intercellular sig-

naling in multicellular environments. Since the ODE part does not lead to a

large discretization system like the PDE part, it is not clear if a splitting method35

can outperform a coupled approach.

In this context, the scope of our work is to present a systematic comparison be-

tween coupled and decoupled approaches for this class of problems. The method

is based on a sensitivity analysis to compute the strength of the coupling. Ad-

ditionally, we compare a multigrid method in which the coupling is considered40

2

only at the coarsest level to a fully coupled approach.

We focus on the solution of local problems. In our test cases we consider

systems with eight cells. Therefore, this solution process can be used for example

as local solver for nonlinear preconditioner of Newton-type methods [16] or

domain decomposition methods [17].45

Application. We solve a coupled PDE/ODE system describing Interleukin-2

(IL-2) signaling between different types of T cells. The underlying model has

been derived by Busse et al. [5] to study immune regulation. The resulting

system consists of one PDE for the intercellular area and of three ODEs for

each biological cell. The coupling occurs in linear and nonlinear terms of the50

equations, as well as in the Robin type boundary conditions. This system is

prototypical for all signaling models for intercellular communication by diffusing

messengers. Other coupled PDE/ODE models for cellular signaling describe

the interaction with multiple diffusing proteins and more detailed intracellular

processes [4] or allow for receptor gradients on the cell surface [2, 3].55

Busse et al. [5] performed two-dimensional (2D) simulations to analyze the

dynamics and pattern formation of intercellular signaling between 170 cells.

The solution of this model in the three-dimensional (3D) case is much more

involved and requires numerical treatment that goes beyond the method used in

the cited work. Therefore, we have developed the numerical method, presented60

in this paper, to efficiently solve the model in realistic 3D environments with

up to 5000 cells [18]. The numerical results have shown that 3D effects have an

important influence on the cell interaction. Thus, new insights into the study

of the immune response have been gained.

Outline. The paper is organized as follows. In section 2 we give an abstract65

description of the model. We present the mathematical formulation and the

functional setting. We discretize the coupled PDE/ODE system by the finite

element method (FEM) in section 3. We use a sensitivity approach in section

4 to analyze the coupling of the PDE/ODE system and in section 5 we present

different solving approaches for the coupled system. We present the numerical70

3

results exemplary for a particular application and discuss numerical aspects in

section 6.

2. Mathematical Models for Intercellular Signaling

A model for intercellular signaling consists of a PDE equation for the inter-

action between the cells in the intercellular area Ω coupled with ODEs for the75

intracellular processes. We denote by Nc the number of cells in Ω and indicate

by Γi the boundary of each cell i for i = 1, . . . , Nc. The outer boundary of Ω is

denoted by Γout.

(a) 8 interacting cells with surfaces Γi,different colors for different cell types

(b) Intercellular area Ω(cutting plane through intercellular area)

Figure 1: Visualization of the computational domain

Depending on the type of intercellular signaling, different nonlinear operators

describe the dynamics in the intercellular area (AΩ), e.g. degradation, the

dynamics on the cell surfaces (AΓi) of each cell and the intracellular processes

(Bi). We denote the solution of the PDE part by u and the vector of solutions

4

of the ODE part by v. A general formulation of the model can be written as

∂tu(t, x)−D∆u(t, x) +AΩ(u(t, x)) = 0 for (t, x) ∈ (0, T ]× Ω,

D∂nu(t, x)−AΓi(u(t, x), vi(t)) = 0 for (t, x) ∈ (0, T ]× Γi,

D∂nu(t, x) = 0 for (t, x) ∈ (0, T ]× Γout,

∂tvi(t) + Bi(ui(t), vi(t)) = 0 for t ∈ (0, T ],

(1)

with given diffusion coefficient D and initial values u(0, x) = u0, v(0) = v0. We

denote the average of u on the surface of Γi by ui and by vi the associated ODE

values with this cell.

ui(t) =

∫Γiu(t, s) ds

|Γi|(2)

Similar systems arise in different applications, e.g. in closed-loop cardiovascular

simulations where the PDE part is coupled with the ODE part on the surfaces80

for in- and outflow, see Moghadam et al. [10].

Remark 2.1. To study the dynamical process and validate the model we com-

pute the entire trajectory. Nevertheless, the simulations mostly converge to a

stable steady state. Therefore we consider the implications on a coupled solver

for a computation of the steady state in section 6.3.1 as well.85

3. Discretization

For a variational formulation we introduce the Hilbert space V p, e.g. V p =

H1, for the PDE part of the equation and the vector space V o = Rn, where n

denotes the number of ordinary differential equations in the system. We define

the product space V := V p × V o.90

We consider the implicit Euler method as time stepping scheme, and dis-

cretize spatially the computational domain Ω by continuous finite elements.

Remark 3.1. For practical use one applies higher order time marching schemes,

e.g. Crank-Nicolson would be a proper choice for the PDE part of the system.

5

The ODE part could be solved by higher order Runge-Kutta or backward differen-95

tiation formulas. To simplify the implementation we choose the unconditionally

stable implicit Euler method for both parts. For the goal of this work, the nu-

merical results regarding the strength of the PDE/ODE coupling are even more

meaningful for higher order schemes as well, see Remark 6.2.

Considering a time step k we use the semi-discretized weak formulation of

the equations (1) to compute (un+1, vn+1) ∈ V in each time step for all ϕ ∈ V p:

(un+1, ϕ)Ω + kD(∇un+1,∇ϕ) + k(AΩ(un+1), ϕ)Ω

+k∑i≤NC

(AΓi(un+1, vn+1

i ), ϕ)Γi = (un, ϕ)Ω,

vn+1 + k∑i≤NC

Bi(un+1i , vn+1

i ) = vn.

(3)

To discretize spatially the system (3), we define a grid Tl0≤l≤L, which100

consists of a subdivision of the domain in quadrilaterals cells K. The diameters

of the cells hK define a mesh parameter h by the piece-wise constant function

h|K = hK . The discrete solution component uh is sought in the finite dimen-

sional space V ph ⊂ V p. We choose V ph as the space of Q1-elements, the space

of functions obtained by transformations of bilinear polynomials defined on a105

reference regular unit cell.

Only the PDE part needs to be discretized by the FEM, but due to the

coupled system the ODE part of the discretized solution vh ∈ V o depends on

the spatial discretization as well. Then we can write the fully discretized version

of system (3) for all ϕh ∈ V ph as follows:

(un+1h , ϕh)Ω + kD(∇un+1

h ,∇ϕh)k + (AΩ(un+1h ), ϕh)Ω

+k∑i≤NC

(AΓi(un+1h,i , v

n+1h,i ), ϕh)Γi

= (unh, ϕh)Ω,

vn+1h + k

∑i≤NC

Bi(un+1h,i , v

n+1h,i ) = vnh .

(4)

A nonlinear coupled system arises as discrete system (4) and needs to be

6

solved for each step of the time marching scheme. For the resulting discrete

nonlinear coupled system we introduce the shorter notation

Ah(u, v) = fh,

Bh(u, v) = gh.(4′)

We use the subscript h also for the operator Bh to indicate its dependence on

the spatial mesh discretization through the coupling with the PDE part. We

omit the subscript h for the solution components u and v to simply the notation

in the next sections.110

4. Sensitivity Analysis of the Coupled System

A splitting solving scheme allows solving both parts of the system with a

solver at hand, possibly tuned to solve that specific part of the problem. The

coupling can be implemented in an external (to the two solvers) framework

allowing an easy implementation of the more complex coupled problem.115

Let Sh : v 7→ u and Th : u 7→ v denote the solution operator for the decoupled

PDE part and ODE part of the discretized system of equations (4′)respectively.

The first equation, u = Sh(v), is solved for a given value of v, then the second

equation, v = Th(u), is solved with the resulting value of u and the cycle is

iterated until a given tolerance is reached. This process can also be written as

a composition of the two operators:

un+1 = Sh(Th(un)

). (5)

A fixed point iteration to solve the system (4′) has a slow convergence rate

(typically only linear) and the number of fixed point iterations depends on the

nature of the coupling and the model parameters. Nevertheless, a splitting linear

solver can be considered advantageous as part of a Newton scheme. Thus, in-

stead of solving the update of the solution using the Jacobian of the full system,120

one updates iteratively the two decoupled parts. In the following sections we

7

will present these two solution approaches, based on a (quasi) Newton method,

in which we compare two different strategies to solve the Newton update.

In the rest of this section, we present a sensitivity approach to decide whether

a fixed point iteration or the full system update should be used. As we show125

later, the choice depends on the actual model parameter and the method gives

a quantitative index that can be used for practical implementations. We use

the well known fact that a fixed point iteration can converge only if a specific

condition on the iteration operator is fulfilled.

Considering the formulation (5) we write the Jacobian of the fixed point

operator as

J =∂Sh∂v

∂Th∂u

(6)

According to the Banach fixed point theorem the following criterion has to

be fulfilled for the convergence of the fixed point iteration

‖J‖ < 1, (7)

in some norm ‖ · ‖. A more convenient criterion is the substitution of the norm

with the spectral radius of the matrix J :

|λmax(J)| < 1. (8)

If this condition is fulfilled a simple fixed point approach converges, therefore130

this criterion has been used, e.g. in [19], to define whether the coupling of

the system (4′) is strong. As will be shown in section 6.3.1, the considered

PDE/ODE system for intercellular signaling is strongly coupled and thus a full

coupling is more effective than a decoupled method. Decoupled methods can

still converge for strong coupled systems if embedded in a Newton’s scheme but135

require a large number of fixed point iterations, as we show in our numerical

results.

We proceed now with the calculation of the largest eigenvalue of the Jacobian

8

J . Therefore we differentiate the discretized operators Ah and Bh and obtain

the sensitivity equations

A′h,u(u, v)uδv +A′h,v(u, v)δv = 0, ∀δv ∈ V o, (9)

and

B′h,v(u, v)vδu +B′h,u(u, v)δu = 0, ∀δu ∈ V ph . (10)

where we have used the notation

uδv :=∂u

∂v(δv), vδu =

∂v

∂u(δu)

for the sensitivities and we have written A′h,u and A′h,v for the derivatives of Ah

with respect to u and v and analogously B′h,u and B′h,v for the derivatives of

Bh. In the decoupled system, uδv indicates the variation of the PDE solution140

perturbing the solution of the ODE system and equivalently vδu is the variation

of the ODE system for a perturbation of the PDE system.

Since the sensitivities in the linear solver strongly depend on the used time

stepping scheme, we consider only the sensitivities for a computation of the

steady state. Then, the equations (9) are stationary PDEs to be solved for145

each component of δv, while the ODE part (10) consists of algebraic equations

solved for each δu. Therefore we compute the sensitivity matrices ∂Sh/∂v as

a No × Np matrix and ∂Th/∂u as a Np × No matrix, where No denotes the

number of ODE equations and Np the dimension of the PDE discretization.

Remark 4.1. In the PDE/ODE system, which we present in section 6, the150

coupling between the two parts appears only at the boundaries Γi and only with

the first two components of v. Thus, the product ∂Sh/∂v ∂Th/∂u decouples into

a block diagonal matrix consisting of 2 × 2 matrices for each biological cell. In

addition, we only need to calculate the sensitivities (10) for the restriction of

δu on the boundaries Γi, which are nonetheless algebraic equations, so that the155

9

major costs to calculate the sensitivities are given by the PDE part (9).

For nonlinear systems of equations the sensitivity analysis depends on a given

point of linearization (u, v). We compute an approximate numerical solution

of the system (4′) for characteristic values of the parameters and choose the

computed solution as point of linearization.160

5. Numerical Schemes

In this section we present different approaches for a solver of a strongly

coupled PDE/ODE system. Their numerical comparison is presented in section

6.

5.1. Nonlinear solver165

Newton-type methods provide a flexible and reliable framework for nonlinear

problems by solving a series of linear equations. We present both a splitting

and a fully coupled solver of the linearized subproblems.

5.1.1. Fully coupled Newton’s method

To apply Newton’s method we linearize the system and solve in each Newton

step the system:A′h,u(un, vn) A′h,v(un, vn)

B′h,u(un, vn) B′h,v(un, vn)

δun+1

δvn+1

=

fh −Ah(un, vn)

gh −Bh(un, vn)

, (11)

to obtain the Newton updates δun+1 and δvn+1 used to calculate the next170

iterates un+1 = un + δvn and vn+1 = vn + δvn+1.

5.1.2. Decoupled inexact Newton’s method

Next, we consider a splitting solving scheme for the linear systems defined

in each Newton-step. In a typical decoupled scheme the two systems are solved

iteratively in separate solvers. In each Newton step the coupled system is solved175

by this decoupled scheme until the residual of the system fulfills a certain accu-

racy or a maximum of iterations is reached.

10

A standard algorithm for a decoupled Newton’s method is shown in Al-

gorithm 1 which iterates until the approximated solution fulfills a prescribed

accuracy (TOLnewton). In this scheme we solve in each time step m with a

Newton-type method the solution for the next time step (un+1, vn+1) by calcu-

lating a few iterations of the decoupled subsystems. For each Newton step n

the decoupled system is solved by the following fixed point iterationA′h,u(un, vn) A′h,v(un, vn)

0 B′h,v(un, vn)

δui+1

δvi+1

=

fh −Ah(un, vn)

gh −Bh(un, vn)−B′h,v(un, vn)δui

(12)

until the Newton updates (δui+1, δvi+1) fulfill the linear residual of the system

(11) to an accuracy (TOLiter).

A common approach to accelerate such a solution process is a Quasi-Newton180

iteration in which the Jacobian matrix is only approximated up to a certain

accuracy. In this way the costs per Newton iteration are reduced, while the

number of Newton iterations increase. A trade-off between accuracy and total

costs can enable a reduction of computing time with respect to a full Newton

method. Such a Quasi-Newton scheme is obtained if a low accuracy (TOLiter)185

or a small maximum number of fixed point iterations (MAXiter) is chosen.

Algorithm 1: Decoupled algorithm: inexact Newton scheme

n = 0repeat

i = 0repeat

compute Newton updates (δui+1, δvi+1) by solving (12)evaluate the residual resiter of the linear system (11)i = i+ 1

until resiter < TOLiter or i = MAXiter

update the iterate un and vn by δun+1 and δvn+1

evaluate the residual resnewton of the nonlinear system (4′)n = n+ 1

until resnewton < TOLnewton

11

This decoupled method is compared to the fully coupled Newton method for

different parameters in numerical tests of section 6.3.1.

5.2. Multigrid scheme

In this section we introduce a multilevel preconditioner which can cope with190

the strong coupling between PDE and ODEs. Such a coupling arises in the

solver of the linear subsystems if the fully coupled Newton method is used

instead of a splitting scheme. Coupled problems are commonly preconditioned

by block preconditioning approaches, e.g. by simple block diagonal methods

or a preconditioning of the Schur complement [20]. We will not use a block195

preconditioning approach because of the small dimension of the ODE part but

instead set up a coupled preconditioner based on the linear multigrid method.

In fact, it is well known [21, 8] that the most efficient preconditioner for the

PDE block is a multilevel preconditioner. In general the number of iterations

of the preconditioned linear solver is independent of the mesh refinement. We

compare this approach to the previously described decoupled solving scheme

in Figure 2. We consider a hierarchy of meshes Tl0≤l≤L, where the index

0 denotes the root mesh, i.e. the coarsest mesh from which all other meshes

are derived by refinement. In this section we use the following notation for the

system matrix of (11)

Kl :=

A′,lh,u A′,lh,v

B′h,u B′h,v

(13)

where the index l indicates the grid refinement level. Note, that we do not use

the notation with superscript l in the blocks of the ODE part. In fact, B′h,v does

not depend on mesh level, while B′h,u does depend on the mesh level through the200

coupling term uh on the cell boundary, but we use the following approximation:

To reduce the computational costs, and simplify the implementation, the cou-

pling ODE/PDE block is calculated at each level with the term uh computed

at the finest level. In this way the whole ODE part does not depend on the

refinement level l. We have observed that this modification does not influence205

12

considerably the performance of the multilevel algorithm.

The multigrid scheme is used as a preconditioner for a Krylov method applied

to the system matrix Kl. We use a generalized minimal residual (GMRES)

method because of the asymmetry of the system matrix, but a different Krylov210

method as, e.g., the BiCG or BiCGStab would also be appropriate for our

purpose.

We show numerically in section 6.3.1 that the efficiency of the preconditioner

is independent of the mesh size. The work presented here is based for the PDE

part on previous work by Janssen and Kanschat [22], while for the coupling215

PDE/ODE we present some new results.

5.2.1. Transfer operators

For the transfer operators we use the following notation

Rl−1l : Vl → Vl−1 (restriction), P l−1

l : Vl−1 → Vl (prolongation). (14)

The restriction and prolongation operators act only on the PDE part, i.e. the

finite element discretization, while the ODE part is transferred by the identity

in both directions. The restriction and prolongation for the PDE part are imple-

mented as intergrid transfers induced by the natural embedding of hierarchical

meshes [22]. In matrix notation the restriction of the whole residual is given by

application of the operator Rl−1l 0

0 I

, (15)

where I ∈ Rn×n denotes the identity matrix of the ODE part. The prolongation

of the whole residual is defined analogously.

13

5.2.2. Smoother220

In case of strong coupled problems a common strategy for the smoothing

process is to consider the full coupling only at the coarsest level and to smooth

the two parts separately (decoupled) on the finer levels.

Since in our case the ODE part is small in comparison with the PDE one,

we expect that the marginally more expensive smoothing of the whole coupled225

system at all levels would be efficient given the strong coupling. Therefore, we

compare the two strategies of (1) smoothing the whole system or (2) smoothing

only the PDE part. For this comparison every efficient smoother would be

appropriate, we have chosen the incomplete LU factorization (ILU). The two

smoother are denoted S1 and S2:230

S1: Incomplete factorization (ILU) of the whole matrix Kl (13);

S2: incomplete factorization (ILU) of the PDE block as part of a Block Gauss

Seidel scheme.

6. Numerical Results

In this section, we make a comparison between the different numerical schemes235

presented in the previous section. The following computations were performed

using the C++ library deal.II, see Bangerth et al. [23], with the UMFPACK

library applied as direct solver on the coarse grid level [24]. Further implemen-

tation details can be found in [25].

6.1. Model240

Exemplary, we focus on a model for signaling of Interleukin-2 (IL-2) between

T cells in the lymph node presented by Busse et al. [5]. Interleukin concen-

trations in the intercellular area regulate the type and strength of the immune

response. The model consists of a reaction-diffusion equation describing the

distribution of IL-2 between the T cells in the intercellular area Ω coupled with245

ODEs for the intracellular processes by a Robin boundary condition.

We use the following notation:

14

• u(t, s) : [0, T ]×Ω→ R describes the concentration of IL-2 in the intercel-

lular area.

• Ri(t), Ci(t) and Ei(t) : [0, T ] → R describe the number of IL-2 receptors250

(IL-2R), built IL-2/IL-2R receptor-complexes and internalized complexes

for each of the simulated T cells. The receptors are distributed homoge-

neously on the cell surfaces.

The mathematical model consists of a PDE

∂tu(t, x) = D∆u(t, x)− kdu(t, x) for all (t, x) in (0, T ]× Ω,

D∂nu(t, s) = qi(t, s)− konRi(t)u(t, s) + koffCi(t) for all (t, s) in (0, T ]× Γi,

∂nu(t, x) = 0 for all (t, x) in (0, T ]× Γout,

(16a)

coupled with three ODEs for each T cell

∂tRi(t) = w0i + w1

i

Ci(t)3

K3 + Ci(t)3− konRi(t)ui(t)

− kiRRi(t) + koffCi(t) + krecEi(t) for all cells i = 1, . . . , Nc,

∂tCi(t) = konRi(t)ui(t)− (koff + kiB)Ci(t),

∂tEi(t) = kiBCi(t)− (krec + kdeg)Ei(t),

ui(t) =

∫Γiu(t, s) ds

|Γi|,

(16b)

for given initial conditions u(0), Ri(0), Ci(0) and Ei(0) for all cells i ≤ Nc. The

used parameters are described in Table 4. Details about the biological processes,255

parameters and initial values are presented in [5].

We consider two cell types which share the same receptor dynamics but differ

in the IL-2 secretion rate:

• Secreting T cells, which omit IL-2 with the secretion rate qi = 2500 mol/h,

• Responding T cells with qi = 0.260

15

Remark 6.1. We evaluate this system of equations with the sensitivity analysis

presented in section 4. For different numbers, size and distribution of biological

cells, as well as moderate secretion rates q in the biological range, see [5], there

exists a unique stationary state, thus it is possible to use a stationary solver.

We obtain maximal eigenvalues λ of the sensitivity matrix such that

5 < λ < 100.

Therefore, this analysis shows quantitatively that the interaction between the

PDE and the ODE part of the system of equations is strong.

We consider a test problem for the numerical computations of eight cells equidis-

tantly distributed in a 3D environment. The configuration is displayed in Figure

1: the responding T cells are in white and the secreting T cell is highlighted.265

For this test problem a unique stationary state exists and we obtain a maximal

eigenvalue of the sensitivity matrix λ = 8.88 which indicates a strong coupling

between the PDE and the ODE part.

6.2. Multigrid preconditioner

We compare the different smoothers S1 and S2 in a series of numerical tests270

for a stationary solver of system (16a), such that the comparison is not influenced

by a time stepping scheme. We compute the number of GMRES steps over all

newton steps (Σn) and the average reduction rate (r) of the residual in each

GMRES step. We proceed with the Newton scheme until an accuracy of 10−6 is

reached. Since the number of Newton steps only depends on the coupling, the275

nonlinearity of the equation and the accuracy of the solver, it is independent

of the grid refinement. Each of the Newton steps is solved for an accuracy of

10−11.

We refine the grid globally until the finest grid with 885673 degrees of free-

dom compared to 367 on the coarse grid for the PDE part. Additionally 24280

degrees of freedom of the ODE part are coupled to the PDE part. We set in

both smoothers the number of iterations to three. This parameter does not

16

influences our comparison. Next we apply the two smoothers S1 and S2 in the

multilevel scheme. Hence, we compare two main approaches: smoothing only

the PDE part or smoothing the whole matrix. Since the coupling between the

Table 1: Reduction rates of different preconditioners

MG-S1 (ILU) MG-S2 (Bl.-ILU)

L log10 r Σs log10r Σs2 2.00 46 1.41 693 1.92 51 1.27 774 1.85 54 1.21 815 1.81 54 1.15 84

Notation: Σs GMRES iterationsin all Newton steps

r average reduction rateL refinement level

285

PDE and the ODE part in the system is strong, we expect that a good smoother

acting on the coupled system works better. In fact, it can be observed that the

coupled ILU smoother S1 is 35% more effective than S2 with much higher re-

duction rates. Therefore we choose to apply the smoother S1 in the following

sections.290

6.3. Comparison of coupled and decoupled schemes

In this section we compare the described coupled and decoupled scheme in

different test cases. Hence the two approaches are:

• a Newton-type method in which the linearized coupled system (11) is

solved by a GMRES solver preconditioned by the multigrid method de-295

scribed in section 5.2 with smoother S1,

• a Newton-type method in which the linearized system is solved in a de-

coupled manner by a fixed point iteration defined by the system (12). The

PDE block is solved by a GMRES solver preconditioned by a multigrid

method following the implementation of [22]. The solver of choice for300

the symmetric part would be the conjugate gradient method (CG), but

17

we use instead the GMRES method for a direct comparison. Neverthe-

less, we have observed that, in combination with the preconditioner, both

solvers have similar performance.

To make the schemes comparable we use a Newton-type method with the same305

accuracy of the GMRES solver of TOLiter = 10−11 for both linearized systems.

In this way the number of Newton steps to solve the nonlinear problem is in-

dependent of the approach and we can compare the total number of GMRES

steps to solve for an accuracy of TOLnewton = 10−6.

Figure 2: Schematic representation of the considered coupled and uncoupled schemes

(a) coupled scheme

Newton-method

↓Krylov-solver

↓Multigrid-preconditioner

↓Smoother S1

(b) decoupled scheme

Newton-method

↓Fixed point iteration

Direct solver Krylov-solver

↓Multigrid-preconditioner

↓PDE Smoother

6.3.1. Stationary solver310

We compare the two solvers to compute the steady state solution of the

system (16a). The tests comprehend simulations with biological parameters

that correspond to strong coupling and simulations with artificial parameters

which correspond to a weakly coupled system.

• In the simulations with biological parameters the maximal value of the315

eigenvalues of the sensitivity matrix is λ = 8.88. We expect thus a strong

PDE/ODE coupling and hence that the decoupled approach is far less

effective than the coupled one.

18

• A weakly coupled test case is created artificially by increasing the parame-

ter kd from 0.1 to 1000. The consequent increment of the degradation of u320

diminishes the influence of the uptake of the cells which depends directly

on the components of v. Thus the PDE part is ‘decoupled’ from the ODE

part: the sensitivity analysis yields a maximal eigenvalue of λ = 0.01,

which indicates that the coupling is very weak.

In Table 2 we compare the number of Newton steps (n) and the num-325

ber of total GMRES iterations (Σs) needed to obtain a solution of accuracy

(TOLnewton). In each Newton step the decoupled scheme described in Algo-

rithm 1 is iterated until a residual resiter < TOLiter is reached without a

given maximum for the number of fixed point iterations MAXiter. The aver-

age GMRES iterations per Newton step is denoted by s and the sum over all330

GMRES steps by Σs. We globally refine the coarse grid three times up to a

number of 114929 degrees of freedom.

Table 2: Coupled vs decoupled solver

biological problem λ = 8.88 modified problem λ = 0.01

decoupled coupled decoupled coupledL n s Σs n s Σs n s Σs n s Σs

2 7 748 5236 7 6.6 46 3 11 33 3 7 213 7 921 6444 7 7.3 51 3 11 33 3 7 214 7 957 6699 7 7.7 54 3 11.7 35 3 7 21

Notation: Σs GMRES iterations during all Newton stepsn Newton stepss average GMRES iterations per Newton stepL refinement levelλ largest eigenvalue of the sensitivity matrices

The results show that the number of Newton steps is independent of the

used solving approach due to the high accuracy of the Jacobian, which in the

decoupled method is obtained using a small TOLiter. This accuracy comes at335

great cost for the decoupled solver, the sum of GMRES iterations is signifi-

cantly higher (by a factor of more than 100) for the strong coupled biological

problem. The coupled solving scheme is very efficient both for the strong and

19

weak coupled problem. The multigrid preconditioner of the coupled scheme

reduces the number of GMRES iterations even in the strong coupled problem340

to around seven. The decoupled approach is more competitive for the weak

coupled problem, though the coupled solver is still 40% faster.

6.3.2. Non-stationary solver

Instead of applying a stationary solver we solve in this section the time-

dependent system (4) until a stationary solution is reached (20 hours using a345

dimensional time variable). In each time step a coupled non-linear system has

to be solved. In contrast to the stationary case, the strength of the coupling is

reduced when using small time steps.

We choose the maximum number of fixed point iterations in each Newton

step (MAXiter) between one and four and compare the fully coupled Newton350

method to the decoupled Quasi-Newton scheme.

Table 3: Decoupled and coupled solving of the non-stationary problem

∆t = 0.1 ∆t = 0.01MAXiter Σn Σs Σn Σs

decoupled 1 1393 3375 5566 140162 754 3792 3395 152173 547 4363 2880 211534 448 4775 2871 23658

coupled 356 1799 2868 11299

Notation: Σn Newton steps in all time stepsΣs Krylow iterations in all Newton stepsMAXiter maximum of iterations

per Newton step

In Table 3 we report the number of computed newton steps (Σn) and the

number of computed GMRES steps (Σs) over all time steps. The results are

listed for computations on a once refined spatial grid (2189 degrees of freedom)

with 200 (∆t = 0.1) or 2000 (∆t = 0.01) time steps.355

A higher maximal number of fixed point iterations per Newton step increases

the accuracy of the linear solver and thus reduces the number of Newton steps.

The decoupled solving scheme with a maximum of four fixed point iterations re-

20

sult in near the same number of Newton steps compared to the coupled solving

scheme but with more than twice the number of computed GMRES steps. Nev-360

ertheless, it can be observed that the number of total GMRES steps decreases

with a reduced number of allowed fixed point iterations per Newton step. Thus

more than one fixed point iteration per Newton step should be avoided, if the

Quasi-Newton method is still converging (this condition cannot be asserted a

priori).365

As already remarked, the effectiveness of the decoupled solver depends on

the strength of the coupling and thus on the size of the time step. In fact for

time steps ∆t = 0.1 the coupled solver needs around half of the iterations of

the decoupled solver, while for smaller time steps (∆t = 0.01) the iterations of

the coupled solver are reduced by 20% compared to the decoupled solver with370

MAXiter = 1.

Remark 6.2. The use of a higher order time scheme, e.g. the Crank-Nicolson

scheme, allows for larger time steps, and hence leads to a stronger coupling

during the time integration. Since the coupled solving method is more effective

(even for small time steps) in the implicit Euler scheme, it works even better in375

a higher order scheme.

7. Conclusion and Outlook

In this paper, we considered a coupled nonlinear system consisting of a

parabolic partial differential equation and many ordinary differential equations,

which emerges e.g. in systems biology by modeling intercellular signaling path-380

ways. We presented the numerical treatment for an application in immunology:

the dynamics of cytokine (Interleukine-2) signaling between different types of T

cells in the lymph node, an intercellular signaling pathway which controls the

immune response.

The presented methods are valid for a larger class of signaling pathways385

involving more diffusing signaling molecules and more chemical interactions be-

21

tween them or for intracellular signaling pathways like in [6] or for other appli-

cations modeled by systems of coupled PDE/ODE.

In this paper, we showed in a systematic way a numerical comparison be-

tween coupled and decoupled approaches for a class of models which special390

structure (coupled PDE/ODE system) could be effectively exploited. For nu-

merical tests a subproblem of eight T cells was considered. Moreover, we

used a sensitivity analysis and its numerical implementation to study the cou-

pling/decoupling strategy. This approach, applied to the model for Interleukin-2

signaling, indicated that a coupling strategy is better suited for a biologically395

relevant range of model parameters. We implemented a solution method based

on a Newton-type solver with a multigrid preconditioner and showed that the

coupling strategy used for all levels of the multigrid is advantageous. Depending

on the time step length, up to around 50% of the computing time is saved by a

reduction of the linear solver iterations.400

We remark that discretizations consisting of globally refined space and time

grids have been used for the computations in this publication. As final comment,

we indicate a possible strategy for reducing the computation time additionally

by using local mesh refinement both in space and time. Different time grids for

the PDE and the ODE part allow, depending on the strength of the coupling,405

to decrease the number of time steps for the computationally expensive PDE

part of the model. The essential question is how to choose the two time grids

without decreasing the overall convergence rate of the method. An a posteriori

error estimator for the errors of the PDE and ODE discretization is necessary

to reach a certain accuracy efficiently by iterative adaptive refinement. The use410

of a refinement strategy based on such an error estimator allows to control the

two time grids separately and obtain an optimal time discretization for both

parts of the system. The complex realization of such a method goes beyond the

scope of this paper and is subject of our current research.

415

22

Parameters

Table 4: Biological Parameters, see [5]

Symbol Value Parameter

qi 0− 22000 mol./cell/h IL-2 secretion rateD 36000 µm2/h Diffusion coefficient of IL-2kd 0,1/h Extracellular IL-2 degradationw0i 150 mol./cell/h Antigen stimulated IL-2 receptor expression rate

w1i 3000 mol./cell/h Feedback induced IL-2 receptor expression rate

K 1000 mol./cell Half-saturation constant of feedback expressionkon 111,6 /nM/h IL-2 association rate constant to IL-2 receptorskoff 0,83/h IL-2 dissociation rate constant from IL-2 receptorskiR 0,64/h Internalization rate constant of IL-2 receptorskiC 1,7/h Internalization rate constant of receptor complexeskrec 9/h Recycling rate constant of IL-2 receptorskdeg 5/h Endosomal degradation constant IL-2 receptorsr 5µm Cell radiusd 5µm Cell to cell distance

Acknowledgements

The authors gratefully acknowledge Prof. Thomas Hofer, DKFZ & BioQuant

Heidelberg, for his support, for fruitful discussions and provisioning this concrete

application in immunology.420

T.C. was supported by Deutsche Forschungsgemeinschaft (DFG) through the

project CA 633/2-1.

E.F. and D.G. were supported by the Helmholtz Alliance on Systems Biology

(SB Cancer, Submodule V.7) and D.G. additionally by ViroQuant.

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