Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 94

Fig. 13 Experimental pressure p vs. the internal-radius r i for human aortas at different ages [ 34 ],
compared with the numerical results obtained by means of the present multiscale model. The
shape of the collagen fiber centerline in the reference configuration is defined as in Eq. ( 20 ) with
v ¼ 0. Age-dependent model parameters are summarized in Table 2 , whereas constant
parameters are set equal to: ' p ¼ 14 : 5 nm, ' c ¼ 287 nm, E o ¼ 1 GPa, E ¼ 80 GPa, g ¼ 22 : 5 ;
e o ¼ 0 : 1 ;' kinks ¼ 14 nm, A m ¼ 1 : 41 nm 2 ; T ¼ 310 : 15 K, l ¼ 1, and L o ¼ 5 lm

Due to the previous symmetry assumptions and disregarding any possible end
effect, problem reduces to an axisymmetric generalized plane strain problem. In
the framework of an incremental approach, the incremental elastic equilibrium
solution for such a composite multi-layered thick-walled structure is obtained at
each pressure variation p as briefly summarized in the Appendix , updating at each
step the fiber chord direction t ð q Þ and the fiber strain e F ð q Þ as functions of the
actual macroscale strain field [ 8 ].
In Fig. 13 , p-r i curves numerically obtained via present model are compared
with the available experimental results proposed in [ 34 ]. In agreement with his-
tological observations [ 31 ], the MTU number N has been set equal to 60. Moreover,
since the experimental data have been obtained on excised dead aortas, both lon-
gitudinal pre-stretch (that is, the direct strain e o , see Appendix ) and smooth muscle
tone have been assumed to vanish, whereas geometrical radial pre-stretch effects
have been taken into account referring the fiber morphological parameters to the
experimentally-analyzed undamaged aortic cross-sections at zero-load state [ 36 ].
Proposed numerical results show the effects related to the variation of some
model parameters, with the aim to reproduce the evolution of the tissue mechanics
induced by age. The age-dependent values of model parameters have been
assumed in agreement with the evidences proposed through the recent specialized
literature [ 3 , 34 , 35 , 57 , 65 ], wherein well-established data and results give indi-
cation on the evolution of some histological, geometric and mechanical charac-
teristics in healthy human aortas. In the lack of detailed experimental data,
parameters have been set within well-established physiological ranges, and their
age-dependent variation has been chosen in agreement with available qualitative
evidences,

disregarding possible differences between the thoracic and the

abdominal aortic branches.