Constitutive and Computational Aspects in Tumor Therapies of Multiphasic Brain Tissue - Computer Models in Biomechanics - page 253

Fig. 19.2 Representative elementary volume (REV) with exemplarily displayed micro-structure
of brain tissue and macroscopic multiphasic-multicomponent modeling approach

homogenization procedure of a representative brain-tissue sample is shown in
Fig. 19.2 , where a deformable solid skeleton ϕ S consisting of tissue cells and vas-
cular 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 over-
all 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 with four components initiated by ϕ S , ϕ B and

={ S,B,I }

= β ϕ β with β

ϕ I

={ }

.
In order to account for the local compositions of the homogenized tissue, scalar
volume fractions n α

L,D

d v α / d v are introduced for the immiscible parts of the aggre-
gate. Therein, d v α and d v are the local volume elements of ϕ α and of the overall
aggregate ϕ . Assuming fully saturated conditions, this leads to the well-known sat-
uration condition

=

n α n S n B n I = + + = 1 . (19.1)

α

With the aid of the volume fractions, two different densities can be distinguished
relating the local masses d m α either to d v α or to d v . These are the effective or
realistic density ρ αR and the partial density ρ α n α ρ αR : =

ρ αR d m α / d v α , α d m α / d v. = = (19.2)

= 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 concentra-
tions c m and molar masses M m . For this purpose, the partial densities ρ β have to be
related to the mixture volume of ϕ I . In conclusion, this leads to

Moreover, the sum of the partial densities yields the aggregate density ρ

n I ρ β
I

with ρ β ρ β c m M m ,

=

I =

(19.3)

where ρ I is the partial density of ϕ β within ϕ I . Finally, the effective density of the
interstitial fluid mixture with respect to the overall aggregate yields ρ IR = β ρ I .