Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 97

Therefore, multiscale structural approaches allow to implement numerical
simulations, that are patient-specific not only for the geometric description of tis-
sues and organs, but also in terms of tissue constitutive properties. This opens to the
possibility of numerically investigating the effects of histological and biochemical
rearrangement on the mechanics of an organ, as well as of estimating, in an inverse-
like scheme, the values of histo-mechano-chemical features by means of non-
invasive techniques. Accordingly, among the available modeling approaches for
the analysis of soft tissues, the structural multiscale rationale can be retained as the
most promising for conceiving and developing groundbreaking virtual tools,
allowing to improve diagnosis and to assess customized clinical treatments.

Acknowledgments Authors would like to thank Professor Franco Maceri for valuable sugges-
tions and fruitful discussions on this paper.
This work was developed within the framework of Lagrange Laboratory, a European research
group comprising CNRS, CNR, the Universities of Rome ''Tor Vergata'', Calabria, Cassino,
Pavia, and Salerno, Ecole Polytechnique, University of Montpellier II, ENPC, LCPC, and
ENTPE.

Appendix

The incremental elastic equilibrium solution for the multi-layered aortic cylinder,
comprising N identical media lamellar units (MLUs) and loaded with a uniform
internal pressure increment p, is herein briefly reported. Reference is made to the
linearly elastic solution proposed in [ 67 ], considering the problem as an axisym-
metric generalized plane strain problem, characterized by a given non-vanishing
constant direct strain increment e o along the cylinder axis z. Accordingly, for the
kth MLU, the incremental components of the displacement field in a cylindrical
system of coordinates result in

u q ¼ A 1 q a k þ A 2 q a k þ e o a 1 q þ B k a 2 q 2 ;

ð 36 Þ

u u ¼ B k qz ; ð 37 Þ

u z ¼ e o z :

ð 38 Þ

The non-trivial increments of strain components are

e q ¼ o u q e u ¼ u q c zu ¼ B k q ;

oq ; q ;

ð 39 Þ

and the components of stress increments are

r q ¼ A 1 b q1 q a k 1 þ A 2 b q2 q a k 1 þ e o b q3 þ B k b q4 q ;

ð 40 Þ