Wolfgang Ehlers · Arndt Wagner
Constitutive and Computational Aspects in
Tumour Therapies of Multiphasic Brain Tissue
Stuttgart, Januar 2012
Institute of Applied Mechanics (Civil Engineering), University of Stuttgart,
Pfaffenwaldring 7, 70569 Stuttgart/ Germany
{ehlers,wagner}@mechbau.uni-stuttgart.de
www.mechbau.uni-stuttgart.de/ls2
Abstract The present contribution concerns the constitutive modelling and the numerical simulation
of brain tissue with a specific focus on tumour therapies carried out by so-called convection-enhanced
delivery processes (CED). The multiphasic modelling approach is based on the Theory of Porous Media
(TPM) and proceeds from a volumetric homogenisation of the underlying micro-structure. The brain
tissue model exhibits an elastic solid skeleton (cells and vascular walls), which is perfused by two liquids,
the blood and the interstitial fluid. The latter is treated as a mixture of two components, namely, a
liquid solvent and a dissolved therapeutic solute. The inhomogeneous and anisotropic nature of the
white-matter tracts is considered by a spatial diversification of the permeability tensors, obtained from
Diffusion Tensor Imaging (DTI). Numerically, the strongly coupled solid-liquid-transport problem is
simultaneously approximated in all primary unknowns (uppc-formulation) using mixed finite elements
and solved in a monolithic manner with an implicit time-integration scheme. Within this procedure,
numerical examples of the CED under two- and three-dimensional conditions are discussed.
Keywords Theory of Porous Media · multiphasic brain tissue · convection-enhanced drug delivery
Preprint Series
Stuttgart Research Centre for Simulation Technology (SRC SimTech)
SimTech – Cluster of Excellence
Pfaffenwaldring 7a
70569 Stuttgart
[email protected]
www.simtech.uni-stuttgart.de
Issue No. 2011-31
Constitutive and Computational Aspects in
Tumour Therapies of Multiphasic Brain Tissue
Wolfgang Ehlers and Arndt Wagner
Abstract The present contribution concerns the constitutive modelling and the numerical simulation of brain tissue with a specific focus on tumour therapies carried
out by so-called convection-enhanced delivery processes (CED). The multiphasic
modelling approach is based on the Theory of Porous Media (TPM) and proceeds
from a volumetric homogenisation of the underlying micro-structure. The brain tissue model exhibits an elastic solid skeleton (cells and vascular walls), which is perfused by two liquids, the blood and the interstitial fluid. The latter is treated as
a mixture of two components, namely, a liquid solvent and a dissolved therapeutic solute. The inhomogeneous and anisotropic nature of the white-matter tracts is
considered by a spatial diversification of the permeability tensors, obtained from
Diffusion Tensor Imaging (DTI). Numerically, the strongly coupled solid-liquidtransport problem is simultaneously approximated in all primary unknowns (uppcformulation) using mixed finite elements and solved in a monolithic manner with an
implicit time-integration scheme. Within this procedure, numerical examples of the
CED under two- and three-dimensional conditions are discussed.
1 Introduction and Treatment Options for Brain Tumours
After a medical detection of a brain tumour, the conventional treatment proceeds
from a highly invasive removal of the tumour in combination with radiotherapy or
chemotherapy. However, the healing success is still low, because malignant tumours
are strongly resistant and regrow frequently. Therefore, the development of alternative and effective treatments is subject of current research.
Any suitable therapeutic agent reaching the malignant brain tumour can have the
desired therapeutic outcome. Basically, there are several methods for the application
Wolfgang Ehlers and Arndt Wagner
Institute of Applied Mechanics (CE), University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart,
e-mail: {Wolfgang.Ehlers, Arndt.Wagner}@mechbau.uni-stuttgart.de
1
2
Wolfgang Ehlers and Arndt Wagner
of pharmaceuticals. Within chemotherapies, the most commonly used method is the
intravenous application of drugs leading to a more or less regular distribution by the
blood circulation. Within this method, only a part of the therapeutic agent reaches
the target area. However, in case of brain tumours, the drugs have to pass the bloodbrain barrier (BBB) to enter the tissue. Unfortunately, this is not possible for most
of the commonly used therapeutic macro-molecules in brain-tumour therapies, cf.
Fig. 1 (left).
A possible solution of the delivery problem is a direct insertion of therapeutic
agents into the extra-vascular space in order to bypass the BBB. Herein, two different basic approaches can be distinguished:
• Implantation of release systems:
– providing a constant concentration at the point of implantation
– driving the distribution by diffusion as a result of concentration gradients
• Infusion of interstitial fluid with dissolved therapeutic agents:
– distributing therapeutic agents by both concentration and pressure gradients
– resulting in an extensive spreading of the therapeutic particles, see Fig. 1
(right)
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small gaps,
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0
1
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only
micro-molecules
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1
100
0
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can pass (e. g. oxygen)
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011
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to the brain
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large molecules
1
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to the body
1
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(usually bad) 00
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normalised concentration [c/c0 ]
The latter more promising approach represents the focus of this contribution and is
generally known as convection-enhanced drug delivery (CED) of therapeutics. This
1.00
infusion of a solution
release systems
0.75
0.50
0.25
0.00
0.00
0.04
0.08
0.12
distance from administration site [m]
0.16
Fig. 1 Left: Sketch of limitations and characteristics of an intra-vascular medication. Right: Comparison of extra-vascular medications in the dispersion of therapeutics after the same time.
pioneering method was introduced by Bobo et al. (1994) and clinically established
by researchers from the National Institutes of Health (NIH). Through a small hole
in the skull, a catheter is directly placed in the brain tissue, while the therapeutic
solution is infused by an external medical pump. With this method, large target areas
can be supplied. However, the prediction of the distribution profile is challenging
since the distribution is affected by the complex nature of living brain tissue. Based
on these remarks, the goal of this contribution is the continuum-mechanical and
computational simulation of living brain tissue with application to the description of
the CED process. We hope that a practising surgeon can be pre-operatively assisted
in his decisions by our and comparable numerical studies.
Tumour Therapies of Multiphasic Brain Tissue
3
2 Continuum-Mechanical Modelling of Human Brain Tissue
The continuum-mechanical description of human brain tissue was always an important task of biomechanical studies. In the context of the hydrocephalus problem,
Hakim and Adams (1965) presented an early hypothesis that the occurring effects
can only be described by the interplay of several brain-tissue components. The first
satisfying mathematical approach, assuming the brain as a porous medium containing a viscous fluid in the extracellular space (ECS), was carried out two decades
later by Nagashima et al. (1987) including a simple numerical simulation of a 2dimensional (2-d) slice of the brain. Since then, various singlephasic and biphasic
brain-tissue models (see, e. g., Taylor and Miller, 2004; Franceschini et al., 2006;
Dutta-Roy et al., 2008) have been developed treating the hydrocephalus problem.
For the specific application to CED, models have been proposed by, e. g., Smith and
Humphrey (2007), Chen and Sarntinoranont (2007) or Linninger et al. (2008). However, there are still open questions concerning the coupling effects, the description
of the deformable porous tissue and its entire pore content. To contribute to the solution of these questions, the present approach is based on the comprehensive Theory
of Porous Media (TPM) in order to describe the multiphasic nature of brain tissue
including the brain solid, the interstitial fluid and the vascular system.
2.1 Multiphasic Modelling based on the TPM
The TPM is a macroscopic continuum theory, which is based on the Theory of Mixtures (TM), cf. Bowen (1976), combined with the concept of volume fractions. For
more details on its foundation, cf., e. g., Ehlers (2002, 2009) and citations therein.
2.1.1 Basic Anatomy of the Human Brain Tissue
To obtain an overview of the modelling problem, the underlying structure of human
brain tissue is briefly summarised. Situated in the rigid scull, the brain is surrounded
by the cerebrospinal fluid (CSF), which is also found in the inner ventricles of the
brain hemispheres. The intracellular space (ICS) of the nervous brain tissue consists
of grey matter at the cerebral cortex (neural cell bodies) and of white-matter tracts
(myelinated axons) in the inner regions. The ECS of the nervous brain tissue is filled
with mobile interstitial fluid. Furthermore, the entire brain is crossed by a highly
branched blood-vessel system.
2.1.2 Constituents, Volume Fractions and Densities
In order to enable the numerical simulation of the CED process, a biphasic fourconstituent model is presented in accordance to Wagner and Ehlers (2010). The
4
Wolfgang Ehlers and Arndt Wagner
homogenisation procedure of a representative brain-tissue sample is shown in Fig. 2,
where a deformable solid skeleton ϕ S consisting of tissue cells and vascular walls
with hyperelastic properties is perfused by two mobile but separated liquid phases,
the blood plasma ϕ B and the overall interstitial fluid ϕ I . The overall interstitial fluid
ϕ I is treated as a real chemical mixture of two components, the liquid solvent ϕ L
and the dissolved therapeutic solute ϕ D . This leads to a ternary model ϕ = ∪α ϕ α
with α = {S, B, I} with four components initiated by ϕ S , ϕ B and ϕ I = ∪β ϕ β with
β = {L, D}.
macroscale
R EV
macroscopic
multiphasic-multicomponent model:
dv
tissue cells
vascular system
interstitial fluid
ϕ=
(with therapeutic solutes)
microscale
“homogenised model”
dvS
dvIB
dv
[
ϕα =
α
= ϕS ∪ ϕB ∪ ϕI
= ϕ S ∪ ϕ B ∪ (ϕ L ∪ ϕ D )
concept of volume fractions
Fig. 2 Representative elementary volume (REV) with exemplarily displayed micro-structure of
brain tissue and macroscopic multiphasic-multicomponent modelling approach.
In order to account for the local compositions of the homogenised tissue, scalar
volume fractions nα = dvα /dv are introduced for the immiscible parts of the aggregate. Therein, dvα and dv are the local volume elements of ϕ α and of the overall
aggregate ϕ . Assuming fully saturated conditions, this leads to the well-known saturation condition
(1)
∑α nα = nS + nB + nI = 1.
With the aid of the volume fractions, two different densities can be distinguished
relating the local masses dmα either to dvα or to dv. These are the effective or
realistic density ρ α R and the partial density ρ α = nα ρ α R :
ρ α R = dmα /dvα ,
ρ α = dmα /dv.
(2)
Moreover, the sum of the partial densities yields the aggregate density ρ = ∑α ρ α .
To include the interstitial fluid mixture ϕ I into the overall description, elements
of the TM have to be embedded in the TPM (Ehlers, 2009). Following this, the
amount of matter of ϕ L and ϕ D within ϕ I has to be expressed by molar concentraβ
β
tions cm and molar masses Mm . For this purpose, the partial densities ρ β have to be
related to the mixture volume of ϕ I . In conclusion, this leads to
ρ β = nI ρIβ
β
with
ρIβ = cβm Mmβ ,
(3)
where ρI is the partial density of ϕ β within ϕ I . Finally, the effective density of the
β
interstitial fluid mixture with respect to the overall aggregate yields ρ IR = ∑β ρI .
Tumour Therapies of Multiphasic Brain Tissue
5
2.1.3 Kinematics of Superimposed Continua
Based on the fundamental assumption of superimposed continua, the TPM proceeds
from the idea that any spatial point x of the current configuration is simultaneously
occupied by material points of all constituents. However, each constituent follows
its own motion such that x = χα (Xα , t) with Xα as the reference position of the
respective material point of ϕ α and time t. This leads to the individual velocity fields
′
xα = dχα (Xα , t)/dt. The solid matrix is described by a Lagrangean formulation via
the solid displacement uS = x − XS as the primary kinematic variable, while the
pore-flow of blood and interstitial proceeds from a modified Eulerian setting via the
′
′
seepage velocities wξ = xξ − xS with ξ = {B, I}. Assuming the velocities of the
′
′
liquid solvent and the interstitial fluid to be approximately identical, xL ≈ xI with
′
nL ≈ nI , the pore-diffusion velocity of the therapeutic solute ϕ D reads dD I = xD
′
− xI . This includes the possibility to define a seepage-like velocity wD = dD I + wI .
2.1.4 Balance Relations
The set of governing equations for the numerical treatment within the finite element method consists of the following balance equations, which are obtained from
partial mass and momentum balances (e. g. Ehlers, 2009). Therein, materially incompressible constituents, no mass exchange between the constituents, quasi-static
conditions and a uniform temperature are assumed.
• Concentration balance of the therapeutic agent ϕ D :
′
I D
′
I D
0 = (nI cD
m )S + n cm div (uS )S + div (n cm wD )
(4)
• Volume balance of the overall interstitial fluid:
0 = (nI )′S + div(nI wI ) + nI div (uS )′S
(5)
• Volume balance of the blood plasma ϕ B :
0 = (nB )′S + div(nB wB ) + nB div(uS )′S
(6)
• Momentum balance of the overall aggregate ϕ = ∪α ϕα :
0 = div T + ρ g .
(7)
Therein, T = ∑α Tα is the overall Cauchy stress, while bα = g characterises uniform constant gravitational force.
2.1.5 Constitutive Settings
The presented balance equations need to be completed with admissible constitutive
relations. An exploitation of the entropy inequality (Ehlers, 2002, 2009) yields re-
6
Wolfgang Ehlers and Arndt Wagner
strictions and conditions for the formulation of constitutive equations such as the
principle of effective stresses, viz.,
ξ
TS = TSE − nS p I and Tξ = TE − nξ pξ R I,
(8)
ξ
where I denotes the second-order identity. The extra-stresses TE of the pore liquids
are neglected due to dimensional reasons (e. g. Ehlers, 2002) as well as osmotic pressure contributions. The partial stress of the solid skeleton contains the pore pressure
p = (nB pBR + nI pIR )/(1 − nS). Following this,
T = TSE − p I .
(9)
Therein, TSE is the effective stress governed by the solid deformation. Concerning
the CED problem, use is made of the small-strain (linear) elasticity concept. In addition, although the brain tissue exhibits an inhomogeneous and anisotropic nature
of the white-matter tracts, which strongly influences the pore-fluid and diffusion
properties, it is assumed that a standard linear elasticity law in the Hookean sense is
sufficient for the description of the brain deformation. Thus,
TSE = 2 µ S ε S + λ S (ε S · I) I .
(10)
Therein, ε S is the linear Green-Lagrangean strain, and µ S and λ S are the Lamé
constants.
In this contribution, the main attention is drawn to the flow of the administered
therapeutical including the drug. As a matter of fact, a slow infusion process with a
slight application dose as is applied within the CED causes only small deformations
in the solid skeleton (as is seen later in Fig. 7). However, if tumour growth or other
diseases, i. e. hydrocephalus, have to be taken into consideration, the tissue model
can be extended to finite elasticity or to finite viscoelasticity (see e. g. Taylor and
Miller, 2004; Ehlers et al., 2009). In both cases, the inclusion of solid anisotropy is
possible.
Moreover, relations for the filter velocities nξ wξ of the pore fluids and for the
drug diffusion nI cD
m dDI need to be specified. Following the detailed constitutive
modelling process described in Acartürk (2009) and Ehlers (2009), appropriate assumptions for the direct momentum production terms p̂ξ and p̂D have to be postulated and inserted into the respective partial momentum balances. In this regard,
Darcy-like filter laws for the pore liquids are obtained in terms of
n ξ wξ = −
Kξ
(grad pξ R − ρ ξ R g)
γξR
(11)
as well as a Fick-like diffusion law for the therapeutic agent:
D
D
n I cD
m dDI = − D grad cm .
(12)
Tumour Therapies of Multiphasic Brain Tissue
7
In the above equations, γ ξ R is the effective fluid weight and Kξ = γ ξ R KSξ /µ ξ R is
the Darcy permeability, involving the effective dynamic fluid viscosities µ ξ R and the
intrinsic permeabilities KSξ . Since the permeabilities are related to the deformation
of the solid skeleton, deformation-dependent intrinsic permeabilities (e. g. Markert,
2007) are included via
nξ nS κ
Sξ
0S
KSξ = K0S
.
(13)
S
ξ
n
n0S
Sξ
Therein, K0S are the permeability tensors of the undeformed solid reference configuration, which need to be equipped with material parameters describing the initial
(anisotropic) intrinsic permeability of the tissue perfusion by the interstitial fluid
SB
(KSI
0S ) and by the blood plasma (K0S ). The exponent κ ≥ 0 governs the non-linear
deformation-dependent behaviour. A possibility for a patient-specific determination
of the coefficients of the permeability tensors and of the effective drug diffusion
tensor DD is described in Section 2.2.
Finally, there is a general need to express the volume fractions of all constituents
either by primary variables and initial conditions or by constitutive equations. In
case of an incompressible solid constituent with an initial volume fraction of nS0S in
the solid reference configuration, nS = nS0S (det FS )−1 holds and simplifies to nS =
nS0S (1 − divuS ) in case of small-strain theories. Thus, as a result of (1), nB and nI
cannot be specified by primary variables individually but only as a sum: nB + nI =
1 − nS . Thus, additional constitutive information is needed. To solve this problem, it
is assumed in the present study that the blood-vessel system is stable and inherent.
Furthermore, since the blood plasma is assumed to incompressible, this leads to the
assumption that nB = nB0S is constant. In particular, it is assumed that nB = nB0S =
0.05, cf. Table 1.
2.2 Inhomogeneous and Anisotropic Perfusion Parameters
Regarding the micro-structural composition of the nervous brain tissue, one can assume that perfusion in grey matter (cell bodies) is isotropic. In contrast, white-matter
perfusion is anisotropic due to a preferred flow direction in the ECS along the axonal
fibres. The micro-structural information of the white-matter tracts can be provided
by Diffusion Tensor Imaging (DTI), cf. Basser et al. (1994). The outstanding feature
of DTI is the possibility to determine the diffusion tensor Dawd of water molecules
in living biological tissue. The symmetric, positive definite apparent water-diffusion
tensor Dnawd of each voxel of the DTI can be written as
Dnawd


 n
0
γ1,awd 0
Dn11 Dn12 Dn13
n
=  Dn21 Dn22 Dn23  ei ⊗ ek =  0 γ2,awd 0  vni ⊗ vni .
n
Dn31 Dn32 Dn33
0
0 γ3,awd

(14)
8
Wolfgang Ehlers and Arndt Wagner
n
Therein, n denotes the voxel number, vni are the eigenvectors and γi,awd
the eigenvaln
ues of Dawd at each voxel. The basic assumption to obtain the required parameters is
that Dawd possesses the same eigenvectors as DD and KSI
0S , as it is proposed by Tuch
et al. (2001). Therefore, a calibration as was shown by Sarntinoranont et al. (2006)
or Linninger et al. (2008) is carried out via
n
D
γi,D
D,n = D̄
n
γi,awd
n
γ̄awd
n
I
and γi,K
I,n = K̄
n
γi,awd
n
γ̄awd
,
(15)
n
is the mean value of the eigenvalues and D̄D and K̄ I are adjusting refwhere γ̄awd
erence values. Thus, the effective drug diffusion tensor DD,n and the anisotropic
permeability tensor KSI,n
0S are computed for each evaluated voxel via
3
DD,n =
n
and KSI,n
D,n (vi ⊗ vi )
∑ γi,D
0S =
i=1
3
n
I,n (vi ⊗ vi ) .
∑ γi,K
(16)
i=1
To show the general feasibility of this procedure, a patient-specific voxel data set is
used here (available at http://www.sci.utah.edu/∼ gk/DTI-data/). In this regard, a custom M ATLAB algorithm was programmed to process the raw binary data.
Due to the irregular distribution of the anisotropic diffusion parameters, cf. Fig. 3
(left), it is not possible to define a closed analytical function for the anisotropic
perfusion parameters. Therefore, the diffusion data is stored in a look-up table and
loaded in a preceding calculation step to provide the full anisotropic perfusion parameters DD for the drug, cf. Fig. 3 (middle), and KSI
0S for the interstitial fluid, respectively.
In order to include micro-structural information of the blood-vessel system, magnetic resonance angiography (MRA) is a promising in vivo approach to locate and
image blood vessels within the brain tissue. In the present study, a blood-vessel
segmentation of a MRA image was carried out using A MIRA, a software platform
allowing for bio-medical data processing. With this tool, it is possible to assign vascular and non-vascular areas by varying blood perfusion parameters KSB
0S , cf. Fig. 3
2
DD
i j [m /s]
high
DD
22
DD
33
DD
12
DD
13
DD
23
e2
e3
low
high
2
DD
i j [m /s]
DD
11
e1
low
Fig. 3 Left: visualisation of diffusion tensors as ellipsoids at a brain slice. Middle: selected values
for all coefficients of the symmetric and anisotropic diffusion tensor DD obtained by DTI. Right:
vascular (grey) and non-vascular (black) areas on a brain slice.
Tumour Therapies of Multiphasic Brain Tissue
9
(right). In this contribution, a microscopical isotropic perfusion is assumed, which
varies in magnitude between vascular and non-vascular regions.
3 Numerical Application
In order to treat the strongly coupled multiphasic and multiphysical problem numerically, the FE solver 1 PANDAS is used. The primary variables of the present initialboundary-value problem (IVBP) are the solid displacement uS with corresponding
test function δ uS associated with the momentum balance (7) of the overall aggregate, the effective pore pressures pξ R with test functions δ pξ R corresponding to the
volume balances (5) and (6) of the interstitial fluid and the blood plasma, and the
D
concentration cD
m with test function δ cm belonging to the concentration balance (4)
of the therapeutic agent. After a transformation of the local balance equations into
weak formulations, the momentum balance of the overall aggregate yields
GuS ≡
Z
T · grad δ uS dv −
Ω
Z
ρ g · δ uS dv −
Z
t · δ uS da = 0 ,
(17)
Γt
Ω
where t = T n is the external stress vector acting on the boundary of the overall
aggregate and n is the outward-oriented unit surface normal. The weak form of the
liquid constituents reads
G pξ ≡
Z
[(nξ )′S + nξ div(uS )′S ] δ pξ R dv −
Ω
−
Z
nξ wξ · gradδ pξ R dv +
Z
v̄ξ δ pξ R da = 0 ,
(18)
Γξ
Ω
v
where v̄ξ = nξ wξ · n is the efflux of liquid volume. Finally, the weak formulation of
the concentration balance is
GcDm ≡
Z
′
I D
′
D
[(nI cD
m )S + n cm div (uS )S ] δ cm dv −
Ω Z
−
Ω
D
n I cD
m wD · grad δ cm dv
+
Z
ı̄D δ cD
m da = 0 ,
(19)
Γı̄D
where ı̄D = nI cD
m wD · n is the molar efflux of the therapeutic agent.
The spatial discretisation of the coupled solid-fluid-transport problem within
a uS -pBR-pIR -cD
m -formulation requires mixed finite elements (see e. g. Ellsiepen,
1999) with a simultaneous approximation of all primary variables. A standard
Galerkin method is applied using extended Taylor-Hood elements with quadratic
1
Porous media Adaptive Nonlinear finite element solver based on Differential Algebraic Systems
(http://www.get-pandas.com)
10
Wolfgang Ehlers and Arndt Wagner
shape functions for uS and linear shape functions for pIR , pBR and cD
m in order
to obtain a stable numerical solution. This leads to a differential-algebraic system
of equations, which is solved in a monolithic manner with an implicit Euler timeintegration scheme.
3.1 Simulation of CED on a Human Brain Slice
As is well known in biomechanics, the in-vivo determination of patient-specific material data is an almost impossible task. Lacking of an alternative, the simulation
parameters are predominantly extracted from the related literature on phantom experiments and numerical studies on animals or gels, see Table 1. However, since the
focus of this contribution is mainly motivated by making available a modelling approach for numerical studies, we do not claim the accuracy for all included material
parameters.
For the present numerical study, a realistic geometry of slice of a human brain
is spatially discretised using 2,100 hexahedral Taylor-Hood elements (one element
in thickness direction). As is shown in Fig. 4, the catheter is virtually placed in the
Table 1 Material parameters for numerical simulations of CED.
Symbol Value
Unit
Description / Reference
ρ IR
0.993 · 10+3
[kg/m3 ]
1.055 · 10+3
[kg/m3 ]
µ IR
µ BR
0.7 · 10−3
3.5 · 10−3
[Ns/m2 ]
[Ns/m2 ]
nI0S
0.20
[-]
nB0S
0.05
[-]
nS0S
0.75
[-]
effective density interstitial fluid (water at 37◦ C) and
effective density of blood plasma according to The Physics
Factbook by Glenn Elert (http://hypertextbook.com)
dynamic viscosity interstitial fluid (water at 37◦ C) and
dynamic viscosity of blood at 37◦ C according to The Physics
Hypertextbook by Glenn Elert (http://physics.info/viscosity)
initial interstitial fluid volume fraction, according to Baxter
and Jain (1989) and citations therein
initial blood volume fraction, according to Baxter and Jain
(1989) and citations therein
initial solidity (cells & vascular walls), remaining term in (1)
µS
1.0 · 10+3
[N/m2 ]
[N/m2 ]
ρ BR
λS
5.0 · 10+3
DD
ij
10−11 - 10−12 [m2 /s]
KiIj
10−7 - 10−8
[m/s]
KiiB
3.0 · 10−3
3.0 · 10−5
1.4
[m/s]
[m/s]
[-]
κ
elastic Lamé constants (E = 2.8 kPa; ν = 0.417), chosen in
magnitude according to Smith and Humphrey (2007); Chen
and Sarntinoranont (2007) and citations therein
order of magnitude of spatial varying drug diffusion coefficient, cf. Section 2.2, according to Baxter and Jain (1989)
and citations therein
order of magnitude of spatial varying Darcy permeability for
the interstitial fluid, cf. Section 2.2, according to Kaczmarek
et al. (1997)
isotropic Darcy permeability coefficient, blood in vascular
and non-vascular regions, based on Su and Payne (2009)
deformation-dependent permeability (assumption)
Tumour Therapies of Multiphasic Brain Tissue
11
catheter position
Fig. 4 Boundary conditions
I
D
for CED on a horizontal brain
v̄ , c0m
solution influx (application dose)
section corresponding to a
spatially fixed
v̄I = 3.33 · 10−7 [m3 /m2 s]
usual application dose. At
(uS = 0)
(=
b Q = 2.5 [ µ l/min])
the brain cortex and at the
inner ventricle, an efflux of
concentration at catheter tip
−3 [mol/l]
interstitial fluid and therapeucD
0m = 3.7 · 10
region of interest
tic agents over the surface is
(ROI)
drained
possible.
brain tissue applying the corresponding boundary conditions for the CED. Furthermore, plane-strain conditions for the brain section are considered in combination
with a horizontally fixed exterior of the brain (cortex). In addition, the blood flow
is neglected in combination with a constant blood volume fraction. Fig. 5 shows the
anisotropic concentration profile in the region of interest (ROI) close to the infusion
point at different time steps (in total three days). The therapeutic agent is distributed
≈ 2 hours
≈ 12 hours
≈ 1 day
≈ 2 days
≈ 3 days
75
50
25
0
D
cD
m /c0m [%]
100
Fig. 5 Anisotropic distribution of the therapeutic agent during a CED process.
as expected in an irregular manner due to the anisotropic permeability parameters.
Moreover, the propagation front is not smooth. The largest value of the interstitial
fluid volume fraction, cf. Fig. 6, is found at the administration site of the catheter,
as the solid constituent is dilated as a result of the infused solution. The excess pressure is a result of the infusion and naturally maximises at the infusion site of the
catheter, whereas the resulting pressure values are still moderate due to the small
80
pIR [N/m2 ]
0.206
nI [ - ]
Fig. 6 Left: Volume fraction
nI of the interstitial fluid.
Right: Interstitial fluid excess
pressure pIR during the infusion process.
0.198
0
application rate. Note that the infusion pressure strongly depends on the infusion
rate, the stiffness of the solid skeleton, and the tissue permeabilities. Note again that
the present example is rather a numerical study than an approach on the basis of
secure and patient-specific data. Nevertheless, numerical studies are important and
provide the basis for a variety of computational results. For example, a decrease in
12
Wolfgang Ehlers and Arndt Wagner
the permeability parameters of the interstitial fluid would result in a faster increase
of the infusion pressure, cf. Fig 7. This is an important aspect in the strategy of a
clinical intervention, as a critical local pressure at the infusion site could lead to
life-threatening effects due to large local dilatations of the tissue.
Fig. 7 Interstitial fluid excess
pressure pIR (left) and norm
of the solid displacements
|uS | at varying permeabilities.
Therein, K I indicates the order of magnitude of the spatial
varying Darcy permeability.
kN
pIR [ m
2]
4.0
2.0
0.0
|uS | [mm]
1.0
K I = 10−7
K I = 10−8
K I = 10−9
24
48
K I = 10−7
K I = 10−8
K I = 10−9
0.5
t [h]
0.0
24
48
t [h]
3.2 Investigations on a Human Brain Hemisphere
For the sake of clarity, the simulation of a CED process was evaluated and discussed
on a 2-d brain slice. But since all formulations are derived in 3 dimensions, realistic
geometries (e. g. of the human brain hemisphere) can be investigated. Therefore, a
standard human brain (commercially available at www.anatomium.com) is used
to survey the general physical behaviour. The geometry was first adapted using the
CAD software R HINOCEROS to prepare the cerebral hemisphere for the spatial discretisation, cf. Fig. 8 (left). Afterwards, the geometry was exported into the mesh
Fig. 8 Left: Geometrical
preparation of the cerebral
hemisphere for the finiteelement meshing. Right:
Numerical simulation results
of a CED process, showing
the spatial therapeutic agent
distribution.
generation toolkit C UBIT and spatial discretised using tetrahedral Taylor-Hood elements. In the vicinity of the incorporated catheter, a finer grid was chosen. Boundary
conditions and material parameters are applied according to Section 3.1. Realised
numerical simulations show the 3-d distribution front of therapeutic agents in the
3-d brain model, cf. Fig. 8 (right).
Tumour Therapies of Multiphasic Brain Tissue
13
4 Summary and Outlook
An appropriate constitutive model based on the TPM was presented, which is able to
capture the infusion process of therapeutic agents into the brain tissue. All existing
physical constituents are included in the modelling approach and a reasonable consideration of anisotropies and heterogeneities of the white-matter tracts was carried
out, as this influences the observed irregular distribution of the infused therapeutic
agents. The investigated model is able to describe the physical effects in a qualitative correct manner. If profound material parameters can be found in the future (by
appropriate clinical studies), the practising surgeons will benefit from preoperative
studies, predicting the distribution of infused therapeutic agents.
Acknowledgements The diffusion tensor MRI brain dataset was obtained by courtesy of G.
Kindlmann (Scientific Computing and Imaging Institute, University of Utah) and A. Alexander
(W. M. Keck Laboratory for Functional Brain Imaging and Behavior, University of WisconsinMadison). Furthermore, proofreading by Dr. Nils Karajan is gratefully acknowledged.
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