J. Biomedical Science and Engineering, 2013, 6, 1-5 JBiSE
http://dx.doi.org/10.4236/jbise.2013.612A001 Published Online December 2013 (http://www.scirp.org/journal/jbise/)
Modeling and simulation in tissue biomechanics: Modern
tools to face an ancient challenge
Giuseppe Vairo
Department of Civil Engineering and Computer Science (DICII), University of Rome “Tor Vergata”, Rome, Italy
Email: vairo@ing.uniroma2.it
Received 1 November 2013; revised 1 December 2013; accepted 16 December 2013
Copyright © 2013 Giuseppe Vairo. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Keywords: History of Biomechanics; Soft Tissue
Biomechanics; Multiscale Homogenization Approaches;
Collagen Bio-Structures
Coinage of the word “Biomechanics” is almost recent.
The first to use this word was Nikolai Bernstein (1896-
1966) by composing the Ancient Greek words βίος (bios,
that is life) and μηχανική (mēchanikē, that is mechanics),
aiming to address the study of the mechanical principles
that regulate living organisms and biological structures.
But origins of the Biomechanics can be retained really
The Edwin Smith surgical papyrus, an Egyptian docu-
ment written in the 17th century BC, described the dif-
ference among cervical sprain, fracture, and fracture-
dislocation. By the time of Hippocrates (4th century BC),
physical means such as traction or local pressure were
being used to correct spinal deformities, as preliminary
clinical treatments based on the first traces of spinal
biomechanics [1]. Aristotle (4th century BC), starting
from the deductive reasoning and the mathematical rea-
soning identified by Socrates and Plato as fundamental
scientific tools, wrote De Motu Animalium (namely, On
the Movement of Animals), wherein he described ani-
mals’ bodies as complex mechanical systems. With the
fall of Greece and the rise of the Roman Empire, natural
philosophy waned in favor of technology. The second
century anatomist, Galen, physician to the Roman em-
peror Marcus Aurelius, wrote a monumental work, On
the Function of the Parts (referring to the parts of the
human body). Perhaps, it can be considered as the pro-
drome of a textbook on Biomechanics. Nevertheless, the
Renaissance produced the first serious attempts at for-
mally and systematically describing biomechanical con-
cepts, as well as at using physical laws and/or materials
sciences for understanding the mechanics of many bio-
logical systems.
Leonardo da Vinci (1452-1519) studied anatomy in the
context of mechanics, analyzed muscle forces as acting
along lines connecting origins and insertions, investi-
gated joint function, as well as addressed spine stability.
Galileo Galilei (1564-1642) was particularly attracted by
bone mechanics and basic principles of allometry. He
observed that animals’ masses increased disproportion-
ately to their size, and their bones must consequently
also disproportionately increase in girth, adapting to
load-bearing rather than mere size. Moreover, he sought
to formulate physical laws by mathematical approaches,
further freeing scientific conclusions from the misper-
ceptions of the senses. The first comprehensive treatise
on Biomechanics, De Motu Animalium, was written by
Giovanni Alfonso Borelli (1608-1679). It contained the
first analysis of weight bearing by the spine, as well as
the first explanation that the levers of the musculoskele-
tal system magnify motion rather than force, so that mus-
cles must produce much larger forces than those resisting
the motion. Borrelli, Marcello Malpighi (1628-1694, the
much younger chair of theoretical medicine) and René
Descartes (1596-1650, mathematician and father of mo-
dern Philosophy) were key thinkers in establishing the
iatrophysical approach to medicine, which held that me-
chanics rather than chemistry was the key to understand-
ing the functioning of the human body [2].
After Borelli, there is little sign of biomechanics in the
literature until the latter half of the 19th century. The idea
of investigating locomotion using cinematography was
first used scientifically by Etienne Marey (1830-1904),
who first correlated ground reaction forces with move-
ment, opening to the modern motion analysis. In Ger-
many, the Weber brothers studied human gait and com-
puted first body segment parameters for human motion
analysis (1836). Their results were later extended by
Christian Wilhelm Braune and his student Otto Fischer
(1889). Relationships between muscle forces and veloc-
ity of contraction were introduced and deepened by Ar-
chibald Vivian Hill (1886-1977), by using a consistent
thermodynamic approach. And starting from the stress
G. Vairo / J. Biomedical Science and Engineering 6 (2013) 1-5
theory of Augustin Cauchy (1789-1857), the railroad
engineer Karl Culmann (1821-1881) and the anatomist
Hermann von Meyer (1801-1869) compared the stress
patterns in a human femur with those in a similarly
shaped crane. Their evidence led Julius Wolff (1836-
1908) to introduce a law of bone remodeling, opening
the door to the modern orthopedic medicine.
These are the origins. This is a brief and incomplete
“genealogy” of Biomechanics, that clearly highlights as
deep roots of Mechanics, Mathematics, Philosophy, Me-
dicine have nourished the big tree of Biomechanics. But,
what is Biomechanics today? And what is the future
trend of Biomechanics?
Biomechanics is surely still the Science that aims to
understand via physical laws and analytical mechanics
many complex phenomena underlying living structures.
It is the Science that attempts to identify factors and
multi-field interactions affecting and/or activating spe-
cific biological processes. But today, in the modern in-
silico era, Biomechanics is also the way for trying to
trace more effective diagnoses, to find solutions and the-
rapies, and to give clinical answers against diseases and
pathological processes. Biomechanics is also the set of
skills and knowledge that allows to design smart medical
devices and materials; that allows to face tissue engi-
neering with the aim of repairing and restoring living
tissues; that enables to develop advanced bio-based tech-
nologies for applications across a wide spectrum of so-
cial needs, in order to perform breakthrough in diagnosis,
treatment and prevention of diseases. Definitively, Bio-
mechanics is a multi-disciplinary framework towards the
enhancing of the environmental health. And this novel
perspective does not overcome the classical concept of
mechanics of life, but it improves and completes the an-
cient challenge aiming at understanding mechanisms that
regulate living matter, intertwining with the growing
advances in science and technology, as well as with the
growing high-quality life expectancy.
In this light, advances in computing technology, nume-
rical methods, and modeling techniques able to account
for multi-field couplings and multi-physics descrip-
tions—which are typical features in biomechanical prob-
lems—have to be continuously pursued, in order to built
up suitable modern tools for facing the renewed chal-
lenges launched by Biomechanics.
As a matter of fact, enhanced modeling formulations
and advanced computational methods open towards the
possibility of developing effective computer-aided simu-
lation tools, allowing reliable, predictive and patient-
specific biomechanical analyses. Moreover, they may
enable to contribute to improving current clinical and
diagnostic approaches in a groundbreaking way. For in-
stance, patient-specific treatments might be conceived
and adapted according to physiopathological responses of
body systems; effects of histological and biochemical re-
arrangement on tissue constituents might be investigated;
and non-invasive techniques for estimating, in an inverse-
like scheme, the values of histo-mechano-chemical fea-
tures might be provided.
In this context, modeling and simulation of soft tissues
can be considered as one of the open and key research
issues [3,4]. In this case, biological constructs can un-
dergo large displacement and/or large strains. Moreover,
they are characterized by a complex non-linear mechan-
ical behavior, strongly depending on tissue histological
features and biological environment. Thus, there is a
great need for the development of accurate constitutive
models accounting for coupled mechanisms, usually oc-
curring at very different length scales, and involving dif-
ferent physics.
Soft tissues are throughout the whole human body and
they include tendons, ligaments, skin, fibrous tissues,
muscles, blood vessels. They link, support, and are part
of other bio-structures and organs, playing a key role in
the biomechanics of many body systems (e.g., muscu-
loskeletal, respiratory, cardiovascular) [5]. Soft 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 sub-
stance, cells and collagen fibers [6].
As confirmed by well-established studies [7], the high-
ly nonlinear constitutive response of soft tissues at the
macroscale is strictly related to the hierarchical organiza-
tion of collagen from nano up to the microscale.
In biological soft tissues, tropocollagen molecules (main-
ly of type I) can be regarded as one-dimensional struc-
tures about 300 nm long and 1 - 2 nm in diameter, made
up of three polypeptide strands, each one being a left-
handed helix. The three helices are twisted together into
a triple helix (namely, a super helix), representing a co-
operative quaternary structure stabilized by covalent
cross-links [5]. A single tropocollagen subunit self-as-
sembles in the extracellular matrix with four other colla-
gen molecules to form units that, in turn, assemble them-
selves into even larger arrays, called fibrils. A collagen
fibril, characterized by a diameter between 50 to 500 nm,
can be thought as a mesoscale structure between mole-
cule at the nanoscale and fiber at the microscale. In turn,
collagen fibrils self-assemble in densely packed tilted
bundles, namely collagen fibers, stabilized by lateral pro-
teoglycans [8]. Within this rigidly organized biostructure,
molecules interact with each other by means of both in-
ter-molecular covalent cross-links (each of them connect-
ing two molecules) and weak bonds (including hydrogen
bonds and other electromagnetic weak interactions), the
former being dominant with reference to the fibril’s elas-
tic behavior [9].
Collagen fibers in soft tissues can be arranged in
Copyright © 2013 SciRes. OPEN ACCESS
G. Vairo / J. Biomedical Science and Engineering 6 (2013) 1-5 3
agreement with a regular (e.g., tendons) or an irregular
(e.g., skin) pattern, and regular tissues (that is, with an
almost regular fiber arrangement) can be conveniently
classified as uni- (e.g., tendons and ligaments) or multi-
(e.g., arterial walls) directional tissues. Accordingly, dif-
ferent collagen patterns and amounts induce different
mechanical responses, namely induce different stiffness
and strength features at the macroscale. As a result, spe-
cific values of compliance for arterial walls in cardio-
vascular system, of stiffness for pulmonary tissues in res-
piratory system, and of extensibility for tendons in mus-
culoskeletal system, are experienced. Moreover, altered
tissue response in disease (e.g., aneurism, keratoconus,
arthofibrosis) arises from pathological tissue remodeling
and arrangement alterations at the nano and microscale,
inducing unphysiological histology and biochemical
composition. Typical ageing/disease-related mechanical
dysfunctions and disorders, such as tissue hyper-exten-
sibility or weakness, that can produce aneurysm patholo-
gies in arterial tissues, can be associated with alterations
at different scales [10-16]: in content of tissue constitu-
ents, in shape of collagen fibers, in collagen genetic pat-
tern, in density of intermolecular cross-links.
Nevertheless, available non-invasive techniques do not
allow to measure directly a number of important histo-
logical, mechanical, and biochemical properties of col-
lagenous tissues, such as, for instance but not exclusively,
collagen content and fiber waviness, collagen cross-link-
ing, elastin amount, and stiffness of elastin networks.
In this scenario, the development of theoretical results
and computational methods for effectively correlating
mechano-regulated physio-pathological processes occur-
ring at very different length scales, as well as to identify
relationships among alterations and diseases, represents
an open challenge at the cutting edge of modern biome-
To date, current modeling approaches widely employ-
ed within advanced computational frameworks allow to
incorporate only some (but not all) histological data, and
include some biochemical effects only through pheno-
menological laws [17-19]. These models are usually
based on parameters having no direct physical or mor-
phological meaning, that have to be suitably tuned in
order to reproduce experimental behavior observed at the
macroscale. Other approaches, namely structural appro-
aches, aim to link model parameters with tissue arrange-
ment and properties at lower scales, either by micro-
macro homogenization techniques or by assuming a suit-
able hyper-elastic macroscopic behavior on the basis of
specific tissue microscale features. Effects related to col-
lagen fibers are generally treated as linearly elastic in the
first case [20-22], whereas geometric nonlinearities (main-
ly related to the fiber crimp) and material nonlinearities
(induced by nanoscale mechanisms within and among
collagen molecules) are often taken into account in hy-
per-elastic models by choosing a suitable representation
of the fiber strain-energy density as in a phenomenologi-
cal approach [18,23,24]. Accordingly, some microscale
features (such as the crimp shape or the thickness of col-
lagen fibers) and any direct relationship with the mole-
cular scale are usually neglected, despite their well-es-
tablished physio-pathological relevance [10-16].
On the other hand, if tissue modeling aims not only to
reproduce the mechanical response at the macroscale but
also, in a patient-specific framework, to contribute for
diagnostic purposes, as well as to describe the tissue re-
sponse evolution induced by physio-pathological altera-
tions in tissue arrangement, each model parameter should
be directly associated with a well-recognizable and mea-
surable property.
This requirement can be satisfied if the tissue struc-
tured hierarchical arrangement is explicitly described,
possibly reducing model complexity by means of mul-
tiscale homogenization techniques. Such an approach,
employed for example in [25,26], is referred to as a
structural multiscale method. It consists in regarding the
tissue at the macroscale as a fiber-reinforced composite
material, wherein properties of reinforcement phase are
recovered by mechanical models at smaller (than the
macro) length scales, coupled each other by means of
consistent inter-scale relationships. Accordingly, the equi-
valent responses of tissue substructures at different scales
can be analytically derived and consistently integrated
and upscaled, allowing to include at the macroscale the
dominant mechanisms occurring at smaller scales. In
some way, the structural multiscale approach exploits the
rationale followed by nature in designing tissues and
building organs.
Structural multiscale elastic constitutive models for
collagen-rich tissues have been recently proposed [3,
26-30], based on homogenization techniques that explic-
itly incorporate nanoscale and microscale mechanisms,
as well as their coupling effects. By describing histo-
logical alterations at nano, micro and macro scales, those
models have been proved to be able to give useful indi-
cations on the deep link between histology and me-
chanical response of both collagenous tissues and body
structures [28-30]. This approach can be generalized, in
the same modeling framework, for including also dam-
age evolution and inelastic effects at different scales,
generally induced by both mechanical and non-mecha-
nical sources [31,32].
In order to account for nano and microscale effects in
macroscale tissue response, a key aspect is represented
by the modeling of collagen behavior at molecular (nano-
scale) and fibril/fiber (microscale) levels, and by the
consistent coupling of these single-scale descriptions.
Nanomechanical response of collagen molecules is
Copyright © 2013 SciRes. OPEN ACCESS
G. Vairo / J. Biomedical Science and Engineering 6 (2013) 1-5
governed by both entropic thermal fluctuation and ener-
getic stretching mechanisms [7,26,33,34], usually address-
ed via numerical atomistic computations [35]. Recently,
an effective modeling approach, simply integrable in a
continuum framework that addresses the equivalent me-
chanical response of the tissue at the macroscale, has
been proposed [3,4,29,30]. It is based on the assumption
that a collagen molecule can be described by an equiva-
lent zero-dimensional elastic nano-structure, wherein en-
tropic and energetic mechanisms act as in series.
Therefore, referring to soft living tissues, structural
multiscale models reveal as really promising for devel-
oping virtual simulation tools, that are efficient from a
computational point of view and are able, at the same
time, to account for patient-specific features, not only in
geometric description of the tissue domains, but also for
the accurate representation of the tissue mechanical prop-
erties. As a result, the effects of changes in histological
arrangement or biochemical environment on the overall
macroscopic functionality of tissues and organs could be
predicted. Thereby, really customized pharmacological
treatments and therapeutic strategies could be conven-
iently designed and applied. Moreover, structural multi-
scale methods based on consistent homogenization pro-
cedures can be successfully integrated with numerical
submodeling techniques, allowing to analyze stress and
strain localization at the microscale, around and within
cells [4,29]. Accordingly, clear quantitative indications
on cellular response to macroscale mechanical stimuli
can be furnished, contributing to understanding activation
and evolution mechanisms of mechano-regulated biolo-
gical processes, and opening also to the possibility to
quantify critical states related to biological transitions
towards pathological conditions. Finally, parametric bio-
mechanical simulations of tissues and organs based on a
multiscale structural framework might be coupled with
non-invasive in-vivo histological and functional meas-
ures. Thereby, following an inverse-like scheme, indirect
estimates of histo-mechano-chemical features, otherwise
unknown, could be furnished by aiming to improve di-
agnostic procedures.
[1] Sanan, A. and Rengachary, S.S. (1996) The history of
spinal biomechanics. Neurosurgery, 39, 657-68.
[2] Porter, R. (1997) The greatest benefit to mankind: A
medical history of humanity from antiquity to the present.
Harper Collins, New York.
[3] Marino, M. and Vairo, G. (2013) Multiscale elastic mod-
els of collagen bio-structures: From cross-linked mole-
cules to soft tissues. In: Gefen, A., Ed., Multiscale Com-
puter Modeling in Biomechanics and Biomedical Engi-
neering. Springer, Berlin-Heidelberg. Studies in Mech-
anobiology, Tissue Engineering and Biomaterials, 14, 73-
102. http://dx.doi.org/10.1007/8415_2012_154
[4] Marino, M. and Vairo, G. (2013) Computational model-
ling of soft tissues and ligaments. In: Jin, Z., Ed., Com-
putational Modelling of Biomechanics and Biotribology
in the Musculoskeletal System: Biomaterials and Tissues,
Woodhead Publishing Series in Biomaterials, 81, Wood-
head Publishing Limited, Cambridge.
[5] van Holde, K.E. and Matthews, C. (1995) Biochemistry.
Publishing Company Inc., Benjamin/Cummings.
[6] Martini, F.H., Timmons, M.J. and Tallitsch, R.B. (1994)
Human anatomy. Prentice Hall, Upper Saddle River.
[7] Fratzl, P. (2008) Collagen: Structure and mechanics.
Springer-Verlag, New York.
[8] Pins, G.D., Christiansen, D.L., Patel, R. and Silver, F.H.
(1997) Self-assembly of collagen fibers. Influence of fi-
brillar alignment and decorin on mechanical properties.
Biophysical Journal, 73, 2164-2172.
[9] Eyre, D.R., Weis, M.A. and Wu, J.J. (2008) Advances in
collagen cross-link analysis. Methods, 45, 65-74.
[10] Bruel, A. and Oxlund, H. (1996) Changes in biomecha-
nical properties, composition of collagen and elastin, and
advanced glycation endproducts of the rat aorta in rela-
tion to age. Atherosclerosis, 127, 155-165
[11] Bruel, A., Ørtoft, G. and Oxlund, H. (1998) Inhibition of
cross-links in collagen is associated with reduced stiff-
ness of the aorta in young rats. Atherosclerosis, 140, 135-
145. http://dx.doi.org/10.1016/S0021-9150(98)00130-0
[12] Mao, J.R. and Bristow, J. (2001) The Ehlers-Danlos syn-
drome: On beyond collagens. Journal of Clinical Inves-
tigation, 107, 1063-1069.
[13] Bailey, A.J. (2001) Molecular mechanisms of ageing in
connective tissues. Mechanisms of Ageing and Develop-
ment, 122, 735-755.
[14] Carmo, M., Colombo, L., Bruno, A., Corsi, F.R.M.,
Roncoroni, L., Cuttin, M.S., Radice, F., Mussini and E.,
Settembrini, P.G. (2002) Alteration of Elastin, Collagen
and their cross-links in abdominal aortic aneurysms. Euro-
pean Journal of Vascular and Endovascular Surgery, 23,
543-549. http://dx.doi.org/10.1053/ejvs.2002.1620
[15] Järvinen, T.A.H., Järvinen, T.L.N., Kannus, P., Jòzsa, L.
and Järvinen, M. (2004) Collagen fibres of the spontane-
ously ruptured human tendons display decreased thick-
ness and crimp angle. Journal of Orthopaedic Research,
22, 1303-1309.
[16] Couppé, C., Hansen, P., Kongsgaard, M., Kovanen, V.,
Suetta, C., Aagaard, P., Kjær, M. and Magnusson, S.P.
(2009) Mechanical properties and collagen cross-linking
of the patellar tendon in old and young men. Journal of
Applied Physiology, 107, 880-886.
Copyright © 2013 SciRes. OPEN ACCESS
G. Vairo / J. Biomedical Science and Engineering 6 (2013) 1-5
Copyright © 2013 SciRes.
[17] Balzani, D., Neff, P., Schröder, J. and Holzapfel, G.A.
(2006) A polyconvex framework for soft biological tis-
sues. Adjustment to experimental data. International
Journal of Solids and Structures, 43, 6052-6070.
[18] Holzapfel, G.A., Gasser, T.C. and Ogden, R.W. (2000) A
new constitutive framework for arterial wall mechanics
and a comparative study of material models. Journal of
Elasticity, 61, 1-48.
[19] Holzapfel, G. and Gasser, T.C. (2001) A viscoelastic
model for fiber-reinforced composites at finite strains:
Continuum basis, computational aspects and applications.
Computer Methods in Applied Mechanics and Engineer-
ing, 190, 4379-4403.
[20] Comninou, M. and Yannas, I.V. (1976) Dependance of
stress-strain nonlinearity of connective tissues on the geo-
metry of collagen fibers. Journal of Biomechanics, 9,
[21] Lanir, Y. (1979) A structural theory for the homogeneous
biaxial stress-strain relationships in flat collagenous tis-
sues. Journal of Biomechanics, 12, 423-436.
[22] Freed, A.D. and Doehring, T.C. (2005) Elastic model for
crimped collagen fibrils. Journal of Biomechanical En-
gineering—ASME, 127, 587-593.
[23] Holzapfel, G.A., Gasser, T.C. and Stadler, M. (2002) A
structural model for the viscoelastic behavior of arterial
walls: Continuum formulation and finite element analysis.
European Journal of Mechanics—A/Solids, 21, 441-463.
[24] Ciarletta, P., Micera, S., Accoto, D. and Dario, P. (2006)
A novel microstructural approach in tendon viscoelastic
modelling at the fibrillar level. Journal of Biomechanics,
39, 2034-2042.
[25] Tang, H., Buehler, M.J. and Moran, B. (2009) A constitu-
tive model of soft tissue: from nanoscale collagen to tis-
sue continuum. Annals of Biomedical Engineering, 37,
1117-1130. http://dx.doi.org/10.1007/s10439-009-9679-0
[26] Maceri, F., Marino, M. and Vairo, G. (2010) A unified
multiscale mechanical model for soft collagenous tissues
with regular fiber arrangement. Journal of Biomechanics,
43, 355-363.
[27] Maceri, F., Marino, M. and Vairo, G. (2010) From cross-
linked collagen molecules to arterial tissue: A nano-mi-
cro-macroscale elastic model, Acta Mechanica Solida
Sinica, 23, 98-108.
[28] Maceri, F., Marino, M. and Vairo, G. (2011) An insight
on multiscale tendon modelling in muscle-tendon inte-
grated behavior. Biomechanics and Modeling in Mech-
anobiology, 11, 505-517.
[29] Marino, M. and Vairo, G. (2012) Stress and strain local-
ization in stretched collagenous tissues via a multiscale
modeling approach. Computer Methods in Biomechanics
and Biomedical Engineering.
[30] Maceri, F., Marino, M. and Vairo, G. (2013) Age-depen-
dent arterial mechanics via a multiscale elastic approach.
International Journal for Computational Methods in En-
gineering Science and Mechanics, 14, 141-151.
[31] Maceri, F., Marino, M. and Vairo, G. (2012) Elasto-
damage modelling of biopolymer molecules response.
Computer Modeling in Engineering and Sciences, 87,
461-482. http://dx.doi.org/10.3970/cmes.2012.087.461
[32] Balzani, D., Brinkhues, S. and Holzapfel, G.A. (2012)
Constitutive framework for the modeling of damage in
collagenous soft tissues with application to arterial walls.
Computer Methods in Applied Mechanics and Engineer-
ing, 213, 139-151.
[33] Bozec, L. and Horton, M. (2005) Topography and me-
chanical properties of single molecules of type I collagen
using atomic force microscopy. Biophysical Journal, 88,
[34] Buehler, M.J. and Wong, S.Y. (2007) Entropic elasticity
controls nanomechanics of single tropocollagen mole-
cules. Biophysical Journal, 93, 37-43.
[35] Buehler, M.J. (2008) Nanomechanics of collagen fibrils
under varying cross-link densities: Atomistic and contin-
uum studies. Journal of the Mechanical Behavior of Bio-
medical Materials, 1, 59-67.