Biomedical Engineering Reference
In-Depth Information
5 What is the Influence of Stiffness Changes of Single Cells
on the Macroscopic Damage Evolution?
The macroscopic model was used to determine the damage evolution in animal
experiments on Brown Norway rats. These experiments are extensively described
in Stekelenburg et al. [
8
] and Loerakker et al. [
15
]. Brown Norway rats were
anesthetized and placed in supine position in an MR compatible loading device
with their left hind limbs fixed. The Tibialis Anterior was mechanically loaded
with an indenter filled with fluid to make it visible on MR images. The indentation
was applied slowly, but kept constant for a period of 2 h and then the load was
relieved. Because an MR compatible system was used it was possible to make high
resolution images before and after indentation. These images were used to create
dedicated finite element models of each individual rat and each loading condition.
T2-weighted imaging was used to measure the location and magnitude of tissue
changes including oedema and tissue damage.
The procedure for the damage evolution in the macro model worked as follows.
In a first step the full indentation as observed in the experiments was applied to the
Finite Element Model. This resulted in an initial, non-uniform strain state in the
loaded. When a strain in an integration point was less than a threshold c
l
there was
no damage. Cells start to be damage when this lower threshold is exceeded. Also
an upper threshold c
h
was defined, based on the RVE studies. Above this threshold
100 % cell damage was assumed. Between the thresholds c
l
and c
h
the percentage
n of cell death was linearly interpolated:
8
<
E
s
\c
l
;
0
if
E
s
c
l
c
l
E
s
c
h
;
n
½¼
ð
1
Þ
c
h
c
l
100
if
:
E
s
[ c
h
;
100
if
Based on the sensitivity analysis with the RVE's the amount of stiffening
around an integration point was calculated with a quadratic function of n. With the
original stiffness c
10
, the current stiffness of the damaged muscle c
10
is defined as:
c
10
¼
fc
10
ð
2
Þ
with:
f
¼
an
2
þ
bn
þ
1
ð
3
Þ
After checking, how many cells were damaged, locally the new stiffness c
10
and
the simulation was repeated until convergence was reached.
Depending on the stiffening factor used for the individual cells the damaged
area changed. For stiffening factors of 2, 3 and 4 fold for the cells the damaged
area in the macroscopic model increased with factors of 1.2, 1.45 and 1.65
respectively.
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