Multiscale Elastic Models of Collagen Bio-structures: From Cross-Linked Molecules to Soft Tissues - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 76

the unrolling mechanisms of the triple helical structure, that tends to disappear
when ' m ' c . In the literature, both linearly [ 8 , 9 ] and non-linearly [ 15 ] elastic
models aiming to describe collagen energetic elasticity can be found.
In biological soft tissues, a single tropocollagen subunit self-assembles in the
extracellular matrix with four other collagen molecules, with regularly staggered
ends, to form units that in turn assemble themselves into even larger arrays, called
fibrils. A collagen fibril, characterized by a diameter between 50 and 500 nm, can
be thought as a mesoscale structure between molecule at the nanoscale and fiber at
the microscale. Within this organized bio-structure, molecules interact each other
by means of both inter-molecular covalent cross-links (each of them connecting
two molecules) and weak bonds (including hydrogen bonds and other electro-
magnetic weak interactions), the former being dominant with reference to the
fibril's elastic behavior [ 17 ].
Recent experimental evidences [ 18 ] revealed a three-dimensional crystallo-
graphic patterns of collagen molecules within fibrils. Nevertheless, simple
arrangement models were proved to be effective in capturing the mechanical key
aspects of fibrils, related to molecules and to their mutual interactions. For
instance, according to the Hodge-Petruska scheme [ 19 ], fibrils can be successfully
modelled as staggered arrays of parallel macromolecules with an axial offset of
about 67 nm and an equilibrium center-to-center distance of about 1.5 nm between
two transversally adjacent molecules.
2.2 Collagen Fibers and Soft Collagenous Tissues
Collagen fibrils are densely packed in bundles called fibers. Adjacent fibrils within
fibers are stabilized by lateral fibril-to-fibril proteoglycan filaments [ 20 ]. As
confirmed by the specialized literature, a controversial matter is if proteoglycans
play or not a significant role in loading transfer among adjacent fibrils. If fibers
consisted in chains of short fibrils interconnected by proteoglycans, then the
among-the-fibrils load-transfer mechanisms would be highly affected by inter-
fibrils links [ 21 ]. Nevertheless, many recent experimental studies [ 22 , 23 ] reveal
extremely few fibril ends, not confirming the thesis of short fibrils. Moreover, other
experimental/numerical results suggest that proteoglycan-based cross-links have a
marginal and unlikely mechanical role in the elastic behavior of a collagen fiber
[ 24 ], although they contribute to fundamental physiological processes. Accord-
ingly, these evidences support the hypothesis that the load-transfer mechanism
among fibrils within fibers is nearly proteoglycan-independent.
Soft collagenous tissues are generally fibrous connective tissues which can be
either dense or loose, depending on the collagen amount; they consist primarily of
elastin, amorphous ground substance, cells and collagen fibers [ 25 ]. Collagen fibers
can be arranged in agreement with a regular (e.g., tendons) or an irregular (e.g.,
skin) pattern, and regular tissues (that is, with a regular fiber arrangement) can be
conveniently classified in uni- (e.g., tendons and ligaments) or multi- (e.g., arterial