169
Biological Sof
7. Biological Soft Tissues
Jay D. Humphrey
A better understanding of many issues of human health, disease, injury, and the treatment
thereof necessitates a detailed quantification of
how biological cells, tissues, and organs respond
to applied loads. Thus, experimental mechanics
can, and must, play a fundamental role in cell biology, physiology, pathophysiology, and clinical
intervention. The goal of this chapter is to discuss
some of the foundations of experimental biomechanics, with particular attention to quantifying
the finite-strain behavior of biological soft tissues in terms of nonlinear constitutive relations.
Towards this end, we review illustrative elastic,
viscoelastic, and poroelastic descriptors of softtissue behavior and the experiments on which they
are based. In addition, we review a new class of
much needed constitutive relations that will help
quantify the growth and remodeling processes
within tissues that are fundamental to long-term
adaptations and responses to disease, injury, and
clinical intervention. We will see that much has
Constitutive Formulations – Overview .... 171
7.2
Traditional Constitutive Relations ..........
7.2.1 Elasticity .....................................
7.2.2 Viscoelasticity ..............................
7.2.3 Poroelasticity
and Mixture Descriptions ..............
7.2.4 Muscle Activation .........................
7.2.5 Thermomechanics ........................
176
177
177
7.3
Growth and Remodeling – A New Frontier
7.3.1 Early Approaches..........................
7.3.2 Kinematic Growth ........................
7.3.3 Constrained Mixture Approach .......
178
178
179
180
7.4
Closure ................................................ 182
7.5
Further Reading ................................... 182
172
172
176
References .................................................. 183
been learned, yet much remains to be discovered
about the wonderfully complex biomechanical
behavior of soft tissues.
there was a need to await the development of a nonlinear theory of material behavior in order to quantify well
that of soft tissues.
It is not altogether surprising, therefore, that biomechanics did not truly come into its own until the
mid-1960s. It is suggested that five independent developments facilitated this: (i) the post World War II
renaissance in nonlinear continuum mechanics [7.2]
established a general foundation needed for developing constitutive relations suitable for soft tissues;
(ii) the development of computers enabled the precisely controlled experimentation [7.3] that was needed
to investigate complex anisotropic behaviors and facilitated the nonlinear regressions [7.4] that were needed
to determine best-fit material parameters from data;
(iii) related to this technological advance, development
of sophisticated numerical methods, particularly the
Part A 7
At least since the time of Galileo Galilei (1564–1642),
there has been a general appreciation that mechanics influences the structure and function of biological
tissues and organs. For example, Galileo studied the
strength of long bones and suggested that they are
hollow as this increases their strength-to-weight ratio
(i. e., it increases the second moment of area given
a fixed amount of tissue, which is fundamental to increasing the bending stiffness). Many other figures in
the storied history of mechanics, including G. Borelli,
R. Hooke, L. Euler, T. Young, J. Poiseuille, and H.
von Helmholtz, contributed much to our growing understanding of biomechanics. Of particular interest herein,
M. Wertheim, a very productive experimental mechanicist of the 19-th century, showed that diverse soft tissues
do not exhibit the linear stress–strain response that is
common to many engineering materials [7.1]. That is,
7.1
170
Part A
Solid Mechanics Topics
Mechanical stimuli
ECM
α β
Cell-cell
G-proteins
Ion
channels
GF
receptor
FAC
changes
Cadherins
Cell
membrane
Mechanobiologic
factors
Cell-matrix
Integrins
Signaling
pathways
CSK
changes
Transcription
factors
Gene
expression
Altered
mechanical
properties
Altered cell
function
Fig. 7.1 Schema of cellular stimuli or inputs – chemical, mechanical, and of course, genetic – and possible mechanobio-
logic responses. The cytoskeleton (CSK) maintains the structural integrity of the cell and interacts with the extracellular
matrix (ECM) through integrins (transmembrane structural proteins) or clusters of integrins (i. e., focal adhesion complexes or FACs). Of course, cells also interact with other cells via special interconnections (e.g., cadherins) and are
stimulated by various molecules, including growth factors (GFs). Although the pathways are not well understood, mechanical stimuli can induce diverse changes in gene expression, which in turn alter cell function and properties as well
as help control the structure and properties of the extracellular matrix. It is clear, therefore, that mechanical stimuli can
affect many different aspects of cell function and thus that of tissues and organs
Part A 7
finite element method [7.5], enabled the solution of
complex boundary value problems associated with soft
tissues, including inverse finite element estimations of
material parameters [7.6]; and (iv) the space race of the
1960s increased our motivation to study the response
of the human body to applied loads, particularly high
G-forces associated with lift-off and reentry as well as
the microgravity environment in space. Finally, it is not
coincidental that the birth of modern biomechanics followed closely (v) the birth of modern biology, often
signaled by discoveries of the structure of proteins, by
L. Pauling, and the structure of DNA, by J. Watson
and F. Crick. These discoveries ushered in the age of
molecular and cellular biology [7.7].
Indeed, in contrast to the general appreciation of
roles of mechanics in biology in the spirit of Galileo,
the last few decades have revealed a fundamentally
more important role of mechanics and mechanical fac-
tors. Experiments since the mid-1970s have revealed
that cells often alter their basic activities (e.g., their expression of particular genes) in response to even subtle
changes in their mechanical environment. For example, in response to increased mechanical stretching,
cells can increase their production of structural proteins [7.8]; in response to mechanical injury, cells can
increase their production of enzymes that degrade the
damaged structural proteins [7.9]; and, in response to
increased flow-induced shear stresses, cells can increase
their production of molecules that change permeability or cause the lumen to dilate and thereby to restore
the shear stress towards its baseline value [7.10]. Cells
that are responsive to changes in their mechanical loading are sometimes referred to as mechanocytes, and
the study of altered cellular activity in response to
altered mechanical loading is called mechanobiology
(Fig. 7.1). Consistent with the mechanistic philosophy
Biological Soft Tissues
of R. Descartes, which motivated many of the early
studies in biomechanics, it is widely accepted that cells
are not only responsive to changes in their mechanical environment, they are also subject to the basic
postulates of mechanics (e.g., balance of linear momen-
7.1 Constitutive Formulations – Overview
171
tum). As a result, basic concepts from mechanics (e.g.,
stress and strain) can be useful in quantifying cellular
responses and there is a strong relationship between
mechanobiology and biomechanics. Herein, however,
we focus primarily on the latter.
7.1 Constitutive Formulations – Overview
There are, of course, three general approaches to
quantify complex mechanical behaviors via constitutive formulations: (i) theoretically, based on precise
information on the microstructure of the material;
(ii) experimentally, based directly on data collected
from particular classes of experiments; and (iii) via
trial and error, by postulating competing relations and
selecting preferred ones based on their ability to fit
data. Although theoretically derived microstructural relations are preferred in principle, efforts ranging from
C. Navier’s attempt to model the behavior of metals to
L. Treloar’s attempt to model the behavior of natural
rubber reveal that that this is very difficult in practice,
particularly for materials with complex microstructures
such as soft tissues. Formulating constitutive relations
directly from experimental data is thus a practical preference [7.11], but again history reveals that it has
been difficult to identify and execute appropriate theoretically based empirical approaches. For this reason,
trial-and-error phenomenological formulations, based
on lessons learned over years of investigation and often
motivated by limited microstructural information, continue to be common in biomechanics just as they are in
more traditional areas of applied mechanics.
Regardless of approach, there are five basic steps
that one must follow in any constitutive formulation:
•
Specifically, the first step is to classify the behavior of the material under conditions of interest, as, for
example, if the material exhibits primarily a fluid-like
or a solid-like response, if the response is dissipative
or not, if it is isotropic or not, if it is isochoric or not,
and so forth. Once sufficient observations enable one to
classify the behavior, one can then establish an appro-
Fibroblast
Endothelial
cell
Collagen
Smooth
muscle
Adventitia
Elastin
Media
Intima
Basal lamina
b)
Fig. 7.2 (a) Schema of the arterial wall, which consists
of three basic layers: the intima, or innermost layer, the
media, or middle layer, and the adventitia, or outermost
layer. Illustrated too are the three primary cells types (endothelial, smooth muscle, and fibroblasts) and two of the
key structural proteins (elastin and collagen). (b) For comparison, see the histological section of an actual artery: the
dark inner line shows the internal elastic lamina, which
separates the thin intima from the media; the lighter shade
in the middle shows the muscle dominated media, and
the darker shade in the outer layer shows the collagendominated adventitia
Part A 7.1
•
•
•
•
delineate general characteristics of the material behavior,
establish an appropriate theoretical framework,
identify specific functional forms of the relations,
calculate values of the material parameters,
evaluate the predictive capability of the final relations.
a)
172
Part A
Solid Mechanics Topics
priate theoretical framework (e.g., a theory of elasticity
within the context of the definition of a simple material
by W. Noll), with suitable restrictions on the possible
constitutive relations (e.g., as required by the Clausius–
Duhem inequality, material frame indifference, and so
forth). Once a theory is available, one can then design appropriate experiments to quantify the material
behavior in terms of specific functional relationships
and best-fit values of the material parameters. Because
of the complex behaviors, multiaxial tests are often preferred, including in-plane biaxial stretching of a planar
specimen, extension and inflation of a cylindrical specimen, extension and torsion of a cylindrical specimen, or
inflation of an axisymmetric membrane, each of which
has a tractable solution to the associated finite-strain
boundary value problem. The final step, of course, is to
ensure that the relation has predictive capability beyond
that used in formulating the relation.
With regard to classifying the mechanical behavior
of biological soft tissues, it is useful to note that they
often consist of multiple cell types embedded within an
extracellular matrix that consists of diverse proteins as
well as proteoglycans (i. e., protein cores with polysaccharide branches) that sequester significant amounts of
water. Figure 7.2 shows constituents in the arterial wall,
an illustrative soft tissue. The primary structural proteins in arteries, as in most soft tissues, are elastin and
different families of collagen. Elastin is perhaps the
most elastic protein, capable of extensions of over 100%
with little dissipation. Moreover, elastin is one of the
most stable proteins in the body, with a normal halflife of decades. Collagen, on the other hand, tends to
be very stiff and not very extensible, with the exception
that it is often undulated in the physiologic state and
thus can undergo large displacements until straightened.
The half-life of collagen varies tremendously depending on the type of tissue, ranging from a few days in the
periodontal ligament to years in bone. Note, too, that
the half-life as well as overall tissue stiffness is modulated in part by extensive covalent cross-links, which
can be enzymatic or nonenzymatic (which occur in diabetes, for example, and contribute to the loss of normal
function of various types of tissues, including arteries).
In general, however, soft tissues can exhibit a nearly
elastic response (e.g., because of elastin) under many
conditions, one that is often nonlinear (due to the gradual recruitment of undulated collagen), anisotropic (due
to different orientations of the different constituents),
and isochoric (due to the high water content) unless
water is exuded or imbibed during the deformation.
Many soft tissues will also creep or stress-relax under
a constant load or a constant extension, respectively.
Hence, as with most materials, one must be careful
to identify the conditions of interest. In the cardiovascular and pulmonary systems, for example, normal
loading is cyclic, and these tissues exhibit a nearly
elastic response under cyclic loading. Conversely, implanted prosthetic devices in the cardiovascular system
(e.g., a coronary stent) may impose a constant distension, and thus induce significant stress relaxation early
on but remodeling over longer periods due to the degradation and synthesis of matrix proteins. Finally, two
distinguishing features of soft tissues are that they typically contain contractile cells (e.g., muscle cells, as in
the heart, or myofibroblasts, as in wounds to the skin)
and that they can repair themselves in response to local
damage. That is, whether we remove them from the
body for testing or not, we must remember that our ultimate interest is in the mechanical properties of tissues
that are living [7.12].
Part A 7.2
7.2 Traditional Constitutive Relations
At the beginning of this section, we reemphasize that
constitutive relations do not describe materials; rather
they describe the behavior of materials under welldefined conditions of interest. A simple case in point is
that we do not have a constitutive relation for water; we
have different constitutive relations for water in its solid
(ice), liquid (water), or gaseous (steam) states, which is
to say for different conditions of temperature and pressure. Hence, it is unreasonable to expect that any single
constitutive relation can, or even should, describe a particular tissue. In other words, we should expect that
diverse constitutive relations will be equally useful for
describing the behavior of individual tissues depending
on the conditions of interest. Many arguments in the
literature over whether a particular tissue is elastic, viscoelastic, poroelastic, etc. could have been avoided if
this simple truth had been embraced.
7.2.1 Elasticity
No biological soft tissue exhibits a truly elastic response, but there are many conditions under which the
Biological Soft Tissues
assumption of elasticity is both reasonable and useful. Toward this end, one of the most interesting, and
experimentally useful, observations with regard to the
behavior of many soft tissues is that they can be preconditioned under cyclic loading. That is, as an excised
tissue is cyclically loaded and unloaded, the stress
versus stretch curves tend to shift rightward, with decreasing hysteresis, until a near-steady-state response
is obtained. Fung [7.12] suggested that this steadystate response could be modeled by separately treating
the loading and unloading curves as nearly elastic; he
coined the term pseudoelastic to remind us that the response is not truly elastic. In practice, however, except
in the case of muscular tissues, the hysteresis is often
small and one can often approximate reasonably well
the mean response between the loading and the unloading responses using a single elastic descriptor, similar
to what is done to describe rubber elasticity. Indeed, although mechanisms underlying the preconditioning of
soft tissues are likely very different from those underlying the Mullin effect in rubber elasticity [7.13], in both
cases initial cyclic loading produces stress softening and
enables one to use the many advances in nonlinear elasticity. This and many other parallels between tissue and
rubber elasticity likely result from the long-chain polymeric microstructure of both classes of materials, thus
these fields can and should borrow ideas from one another (see discussion in [7.14] Chap. 1). For example,
Cauchy membrane stress (g/cm)
120
RV epicardium
equibiaxial stretch
Circumferential
Apex–to–base
60
1
1.16
1.32
Stretch
Fig. 7.3 Representative tension–stretch data taken, fol-
lowing preconditioning, from a primarily collagenous
membrane, the epicardium or covering of the heart, tested
under in-plane equibiaxial extension. Note the strong nonlinear response, anisotropy, and negligible hysteresis over
finite deformations (note: membrane stress is the same as
a stress resultant or tension, thus having units of force per
length though here shown as a mass per length)
advances in rubber elasticity have taught us much about
the importance of universal solutions, common types of
material and structural instabilities, useful experimental
approaches, and so forth [7.14–16]. Nonetheless, common forms of stress–strain relations in rubber elasticity
– for example, neo-Hookean, Mooney–Rivlin, and Ogden – have little utility in soft-tissue biomechanics and
at times can be misleading [7.17]. A final comment with
regard to preconditioning is that, although we desire to
know properties in vivo (literally in the body), it is difficult in practice to perform the requisite measurements
without removing the cells, tissues, or organs from the
body so that boundary conditions can be known. This
process of removing specimens from their native environment necessarily induces a nonphysiological, often
poorly controlled strain history. Because the mechanical
behavior is history dependent, the experimental procedure of preconditioning provides a common, recent
strain history that facilitates comparisons of subsequent
responses from specimen to specimen. For this reason,
the preconditioning protocol should be designed well
and always reported.
Whereas a measured linear stress–strain response
implies a unique functional relationship, the nonlinear, anisotropic stress–strain responses exhibited by
most soft tissues (Fig. 7.3) typically do not suggest
a specific functional relationship. In other words, one
must decide whether the observed characteristic stiffening over finite strains (often from 5% to as much
as 100% strain) is best represented by polynomial,
exponential, or more complex stress–strain relations.
Based on one-dimensional (1-D) extension tests on
a primarily collagenous membrane called the mesentery, which is found in the abdomen, Fung showed
in 1967 that it can be useful to plot stiffness, specifically the change of the first Piola–Kirchhoff stress with
respect to changes in the deformation gradient, versus stress rather than to plot stress versus stretch as is
common [7.12]. Specifically, if P is the 1-D first Piola–
Kirchhoff stress and λ is the associated component of
the deformation gradient (i. e., a stretch ratio), then
seeking a functional form P = P(λ) directly from data
is simplified by interpreting dP/ dλ versus P. For the
mesentery, Fung found a near-linear relation between
stiffness and stress, which in turn suggested directly
(i. e., via the solution of the linear first-order ordinary
differential equation) an exponential stress–stretch relationship (with P(λ = 1) = 0):
α β(λ−1)
dP
e
−1 ,
= α+βP → P =
dλ
β
(7.1)
173
Part A 7.2
0
7.2 Traditional Constitutive Relations
174
Part A
Solid Mechanics Topics
where α and β are material parameters, which can be
determined via nonlinear regressions of stress versus
stretch data or more simply via linear regressions of
stiffness versus stress data. Note, too, that one could
expand and then linearize the exponential function to
relate these parameters to the Young’s modulus of
linearized elasticity if so desired. Albeit a very important finding, a 1-D constitutive relation cannot be
extended to describe the multiaxial behavior that is
common to many soft tissues, ranging from the mesentery to the heart, arteries, skin, cornea, bladder, and
so forth. At this point, therefore, Fung made a bold
hypothesis. Given that this 1-D first Piola–Kirchhoff
versus stretch relationship was exponential, he hypothesized that the behavior of many soft tissues could
be described by an exponential relationship between
the second Piola–Kirchhoff stress tensor S and the
Green strain tensor E, which are conjugate measures
appropriate for large-strain elasticity. In particular, he
suggested a hyperelastic constitutive relation of the
form:
∂W
∂Q
whereby S =
= c eQ
,
W = c eQ − 1
∂E
∂E
(7.2)
Part A 7.2
where W is a stored energy function and Q is a function of E. Over many years, Fung and others suggested,
based on attempts to fit data, that a convenient form
of Q is one that is quadratic in the Green strain, similar to the form of a stored energy function in terms of
the infinitesimal strain in linearized elasticity. Fung argued that, because Q is related directly to ln W, material
symmetry arguments were the same for this exponential stored energy function as they are for linearized
elasticity. Q thus contains two, five, or nine nondimensional material parameters for isotropic, transversely
isotropic, or orthotropic material symmetries, respectively, and this form of W recovers that for linearized
elasticity as a special case. The addition of a single
extra material parameter, c, having units of stress, is
a small price to pay in going from linearized to finite elasticity, and this form of W has been used with
some success to describe data on the biaxial behavior of skin, lung tissue, arteries, heart tissue, urinary
bladder, and various membranes including the pericardium (which covers the heart) and the pleura (which
covers the lungs). In some of these cases, this form
of W was modified easily for incompressibility (e.g.,
by introducing a Lagrange multiplier) or for a twodimensional (2-D) problem. For example, a commonly
used 2-D form in terms of principal Green strains
is
2
2
+ c2 E 22
+ 2c3 E 11 E 22 − 1 ,
W = c exp c1 E 11
(7.3)
where ci are material parameters. It is easy to show
that physically reasonable behavior is ensured by c > 0
and ci > 0 and that convexity is ensured by the additional condition c1 c2 > c23 [7.18, 19]. Nonetheless,
(7.2) and (7.3) are not without limitations. This 2-D
form limits the degree of strain-dependent changes
in anisotropy that are allowed, and experience has
shown that it is typically difficult to find a unique
set of material parameters via nonlinear regressions of
data, particularly in three dimensions. Perhaps more
problematic, experience has also revealed that convergent solutions can be difficult to achieve in finite
element models based on the three-dimensional (3-D)
relation except in cases of axisymmetric geometries.
The later may be related to the recent finding by
Walton and Wilber [7.20] that the 3-D Fung stored energy function is not strongly elliptic. Hence, despite
its past success and usage, there is clearly a need
to explore other constitutive approaches. Moreover,
this is a good reminder that there is a need for
a strong theoretical foundation in all constitutive formulations.
The Green strain is related directly to the right
Cauchy Green tensor C = F F, where F is the deformation gradient tensor, yet most work in finite elasticity
has been based on forms of W that depend on C, thus
allowing the Cauchy stress t to be determined via (as
required by Clausius–Duhem, and consistent with (7.2)
for the relation between the second Piola–Kirchhoff
stress and Green strain):
t=
∂W 2
F
F .
det F ∂C
(7.4)
For example, Spencer [7.21] suggested that materials
consisting of a single family of fibers (i. e., exhibiting
a transversely isotropic material symmetry) could be
described by a W of the form
W = Ŵ(I, II, III, IV, V) ,
(7.5)
where
I = tr C,
2II = (tr C)2 − tr C2 ,
IV = M · CM,
V = M·C M ,
2
III = det C,
(7.6)
and M is a unit vector that identifies the direction of the
fiber family in a reference configuration. Incompress-
Biological Soft Tissues
ibility, III = 1, reduces the number of invariants by one
while introducing an arbitrary Lagrange multiplier p,
namely
p
(7.7)
W = W̃(I, II, IV, V) − (III − 1) ,
2
yet it is still difficult to impossible to rigorously determine specific functional forms directly from data.
Indeed, this problem is more acute in cases of twofiber families, including orthotropy when the families
are orthogonal. Hence, subclasses of this form of
W have been evaluated in biomechanics. For example, Humphrey et al. [7.22] showed, and Sacks and
Chuong [7.23] confirmed, that a stored energy function of the form Ŵ(I, IV), determined directly from
in-plane biaxial tests (with I and IV separately maintained constant) on excised slabs of noncontracting
heart muscle to be a polynomial function, described
well the available in-plane biaxial stretching data. This
1990 paper [7.22] also illustrated the utility of performing nonlinear regressions of stress–stretch data
using data sets that combined results from multiple
biaxial tests as well as the importance of respecting
Baker–Ericksen-type inequalities [7.2] in the parameter
estimations. Holzapfel et al. [7.18] and others have similarly proposed specific forms of W for arteries based on
a subclass of the two-fiber family approach of Spencer.
Specifically, they propose a form of W that combines
that of a neo-Hookean relation with a simple exponential form for two-fiber families, namely
W̃ = c(tr C − 3)
c1 exp b1 (M1 · CM1 − 1)2 − 1
+
b1
c2 exp b2 (M2 · CM2 − 1)2 − 1 ,
+
b2
(7.8)
175
depend on traditional invariants of C is not optimal,
hence alternative invariant sets should be identified and
explored [7.18, 24, 25]. Experiments designed based on
these invariants remain to be performed, however.
Preceding the one- and two-fiber family models were the microstructural models proposed by
Lanir [7.26]. Briefly, electron and light microscopy reveal that the elastin and collagen fibers within many soft
tissues have complex spatial distributions (notable exceptions being collagen fibers within tendons, which are
coaxial, and those within the cornea of the eye, which
are arranged in layered orthogonal networks). Moreover, it appears that, despite extensive cross-linking at
the molecular level, these networks are often loosely
organized. Consequently, Lanir suggested that a stored
energy function for a tissue could be derived in terms
of strain energies for straightened individual fibers if
one accounted for the undulation and distribution of the
different types of fibers, or alternatively that one could
postulate exponential stored energy functions for the
individual fibers (cf. (7.3)) and simply account for distribution functions for each type of fiber. For example,
for a soft tissue consisting primarily of elastin and type I
collagen (i. e., only two types of constituents), one could
consider
Ri (ϕ, θ) wif λif sin ϕ dϕ dθ , (7.9)
φi
W=
i=1,2
φi
where
and Ri are, respectively, the volume fraction
and distribution function for constituent i (elastin or collagen), and wif is the 1-D stored energy function for
a fiber belonging to constituent i. Clearly, the stress
could be computed as in (7.1) provided that the fiber
stretch can be related to the overall strain, which is
easy if one assumes affine deformations. In principle,
the distribution function could be determined from histology and the material parameters for the fiber stored
energy function could be determined from straightforward 1-D tests, thus eliminating, or at least reducing, the
need to find many free parameters via nonlinear regressions in which unique estimates are rare. In practice,
however, it has been difficult to identify the distribution functions directly, thus they have often been
assumed to be Gaussian or a similarly common distribution function. Although proposed as a microstructural
model, the many underlying assumptions render this
approach microstructurally motivated at best; that is,
there is no actual modeling of the complex interactions
(including covalent cross-links, van der Waals forces,
etc.) between the many different proteins and proteoglycans that endow the tissue with its bulk properties,
Part A 7.2
where Mi (i = 1, 2) denote the original directions of
the two-fiber families. Although neither of these forms
is derived directly from precise knowledge of the microstructure nor inferred directly from experimental
data, this form of W was motivated by the idea that
elastin endows an artery with a nearly linearly elastic
(neo-Hookean)-type response whereas the straightening of multiple families of collagen can be modeled by
exponential functions in terms of fiber stretches. Thus
far, this and similar forms of W have proven useful in
large-scale computations and illustrates well the utility
of the third approach to modeling noted above – trial
and error based on experience with other materials or
similar relations. It is important to note, however, that
inferring forms of W for incompressible behaviors that
7.2 Traditional Constitutive Relations
176
Part A
Solid Mechanics Topics
which are measurable using standard procedures. Indeed, perhaps one reason that this approach did not
gain wider usage is that it failed to predict material
behaviors under simple experimental conditions, thus
like competing phenomenological relations it had to
rely on nonlinear regressions to obtain best-fit values
of the associated material parameters. As noted earlier, the extreme complexity of the microstructure of
soft tissues renders it difficult to impossible to derive
truly microstructural relations. Nevertheless, as we discuss below, microstructurally motivated formulations
can be a very useful approach to constitutive modeling provided that the relations are not overinterpreted.
Among others, Bischoff et al. [7.27] have revisited
microstructurally motivated constitutive models with
a goal of melding them with phenomenological models.
In summary, although no soft tissue is truly elastic in its behavior, hyperelastic constitutive relations
have proven useful in many applications. We have reviewed but a few of the many different functional forms
reported in the literature, thus the interested reader
is referred to Fung [7.12], Maurel et al. [7.28], and
Humphrey [7.19] for additional discussion.
7.2.2 Viscoelasticity
Although the response of many soft tissues tends
to be relatively insensitive to changes in strain rate
over physiologic ranges, soft tissues creep under
constant loads and they stress-relax under constant
deformations. Among others, Fung [7.12] suggested
that single-integral heredity models could be useful in
biomechanics just as they are in rubber viscoelasticity [7.29]. For example, we recently showed that sets
of strain-dependent stress relaxation responses of a collagenous membrane, before and after thermal damage,
can be modeled via [7.30]
Part A 7.2
τ
G(τ − s)
t(τ) = − p(τ)I + 2F(τ)
0
∂ ∂W
×
(s) ds F (τ) ,
∂s ∂C
(7.10)
where G is a reduced relaxation function that depends
on fading time (τ − s) and W was taken to be an exponential function similar to (7.3); all other quantities
are the same as before except with explicit dependence
on the current time. Various forms of the reduced relaxation function can be used, including a simple form that
we found to be useful in thermal damage
G(x) =
1− R
+R,
1 + (x/τR )n
(7.11)
where n is a free parameter, τR is a characteristic time
of relaxation, and R is the stress remainder, that is, the
fraction of the elastic response that is left after a long
relaxation (e.g., R = 0 for a viscoelastic fluid).
If short-term responses are important, numerous
models can be used; for example, the simple viscohyperelastic approach of Beatty and Zhou [7.31] is useful
in modeling biomembranes [7.32]. Briefly, the Cauchy
stress is assumed to be of the general form t = t̂(B, D),
where B = FF is the left Cauchy–Green tensor, with
F the deformation gradient, and D = (L + L )/2 is the
stretching tensor, with the velocity gradient L = ḞF−1 .
Specifically, assuming incompressibility, the Cauchy
stress has three contributions: a reaction stress, an elastic part, and a viscous part, namely
t = − pI + 2F
∂ W̃ F + 2μD ,
∂C
(7.12)
where p is again a Lagrange multiplier that enforces
incompressibility, W is the same strain energy function that was used in the elastic-only description noted
above, and μ is a viscosity. Hence, this description
of short-term nonlinear viscohyperelasticity adds but
one additional material parameter to the constitutive
equation, and the elastic response can be quantified
first via quasistatic tests, thereby reducing the number
of parameters in each regression. Some have considered a synthesis of the short- and long-term models.
For a discussion of other viscoelastic models in softtissue mechanics, see Provenzano et al. [7.33] or
Haslach [7.34] and references therein.
7.2.3 Poroelasticity
and Mixture Descriptions
Not only do soft tissues consist of considerable water, every cell in these tissues is within ≈ 50 μm of
a capillary, which is to say close to flowing blood.
Clearly then, it can be advantageous to model tissues as
solid–fluid mixtures under many conditions of interest.
A basic premise of mixture theory [7.2] is that balance
relations hold both for the mixture as a whole and for the
individual constituents, with the requirement that summation of the balance relations for the constituents must
yield the classical relations. Moreover, it is assumed
that the constituent balance relations include additional
constitutive relations, particularly those that model the
Biological Soft Tissues
exchanges of mass, momentum, or energy between constituents. The first, and most often used, approach to
model soft tissues via mixture theory was proposed by
Mow et al. [7.35]. Briefly, their linear biphasic theory
treated cartilage as a porous solid (i. e., the composite
response due to type II collagen, proteoglycans, etc.),
which was assumed to exhibit a linearly elastic isotropic
response, with an associated viscous fluid within. They
proposed constitutive relations for the solid and fluid
stresses of the form
t (s) = −φ(s) pI + λs tr(ε)I + 2μs ε ,
t (f) = −φ(f) pI − 23 μf div v(f) I + 2μf D ,
(7.13)
and, for the momentum exchange between the solid and
fluid,
− p(f) = p(s) = p∇φ(f) + K (v(f) − v(s) ) ,
(7.14)
177
7.2.4 Muscle Activation
Another unique feature of many soft tissues is their
ability to contract via actin–myosin interactions within
specialized cells called myocytes. Examples include
the cardiac muscle of the heart, skeletal muscle of the
arms and legs, and smooth muscle, which is found
in many tissues including the airways, arteries, and
uterus. The most famous equation in muscle mechanics is that postulated in 1938 by A. V. Hill to describe
force–velocity relations. This relation, like many subsequent ones, focuses on 1-D behavior along the long axis
of the myocyte or muscle; data typically comes from
tests on muscle fibers or strips, or in some cases rings
taken from arteries or airways. Although much has been
learned, much remains to be learned particularly with
respect to the multiaxial behavior. The interested reader
is referred to Fung [7.12]. Zahalak et al. [7.42], and
Rachev and Hayashi [7.43]. In addition, however, note
that modeling muscle activity in the heart (i. e., the electromechanics) has advanced significantly and represents
a great example of the synthesis of complex theoretical,
experimental, and computational methods. Toward this
end, the reader is referred to Hunter et al. [7.44, 45].
7.2.5 Thermomechanics
The human body regulates its temperature to remain
within a narrow range, and for this reason there has
been little attention to constitutive relations for thermomechanical behaviors of cells, tissues, and organs.
Nevertheless, advances in laser, microwave, highfrequency ultrasound, and related technologies have
encouraged the development and use of heating devices
to treat diverse diseases and injuries. For example, supraphysiologic temperatures can destroy cells and shrink
collagenous tissue, which can be useful in treating cancer and orthopedic injuries, respectively. Laser-based
corneal reshaping, or LASIK, is another prime example. Due to space limitations here, the interested reader
is referred to Humphrey [7.46], and references therein,
for a brief review of the growing field of biothermomechanics and insight into ways in which experimental
mechanics and constitutive modeling can contribute.
Also see Diller and Ryan [7.47] for information on the
associated bioheat transfer. Of particular note, however,
it has been shown that increased mechanical loading
can delay the rate at which thermal damage accrues,
hence there is a strong thermomechanical coupling and
a pressing need for more mechanics-based studies – first
experimental, then computational.
Part A 7.2
where the superscripts and subscripts ‘s’ and ‘f’ denote solid and fluid constituents, hence v(s) and v(f)
are solid and fluid velocities, respectively. Finally, φ(i)
are constituent fractions, μs and λs are the classical
Lamé constants for the solid, μf is the fluid viscosity, and ε is the linearized strain in the solid. In some
cases the fluid viscosity is neglected, thus allowing tissue viscoelasticity to be accounted for solely via the
momentum exchange between the solid and diffusing
fluid, where K is related to the permeability coefficient.
Mow and colleagues have developed this theory over
the years to account for additional factors, including the
presence of diffusing ions [7.36, 37] (see also [7.38]
for a related approach). Because of the complexity of
poroelastic and mixture theories, as well as the inherent geometric complexities associated with most real
initial–boundary value problems in soft tissues, finite
element methods will continue to prove essential; see,
for example, Spilker et al. [7.39] and Simon et al. [7.40]
for such formulations. In summary, one can now find
many different applications of mixtures in the literature
on soft tissues (e.g., Reynolds and Humphrey [7.41] address capillary blood flow within a tissue using mixture
theory) and, indeed, the past success and future promise
of this approach mandates intensified research in this
area, research that must not simply be application, but
rather should include development and extension of
past theories. Moreover, whereas many of the experiments in biomechanics have consisted of unconfined
or confined uniaxial compression tests using porous indenters, there is a pressing need for new multiaxial
tests.
7.2 Traditional Constitutive Relations
178
Part A
Solid Mechanics Topics
7.3 Growth and Remodeling – A New Frontier
As noted above, it has been thought at least since the
time of Galileo that mechanical stimuli play essential
roles in governing biological structure and function.
Nevertheless an important step in our understanding of
the biomechanics of tissues began with Wolff’s law for
bone remodeling, which was put forth in the late 19-th
century. Briefly, it was observed that the fine structure of cancellous (i. e., trabecular) bone within long
bones tended to follow lines of maximum tension. That
is, it appeared that the stress field dictated, at least
in part, the way in which the microstructure of bone
was organized. This observation led to the concept of
functional adaptation wherein it was thought that bone
functionally adapts so as to achieve maximum strength
with a minimum of material. For a discussion of bone
growth and remodeling, see Fung [7.12], Cowin [7.48],
and Carter and Beaupre [7.49]. Although the general
concept of functional adaptation appears to hold for
most tissues, it is emphasized that bone differs significantly from soft tissues in three important ways. First,
bone growth occurs on surfaces, that is, via appositional
growth rather than via interstitial growth as in most
soft tissues; second, most of the strength of bone derives from an inorganic component, which is not true
in soft tissues; third, bone experiences small strains and
exhibits a nearly linearly elastic, or poroelastic, behavior, which is very different from the nonlinear behavior
exhibited by soft tissues over finite strains. Hence, let
us consider methods that have been applied to soft
tissues.
7.3.1 Early Approaches
Part A 7.3
Murray [7.50] suggested that biological “organization
and adaptation are observed facts, presumably conforming to definite laws because, statistically at least, there
is some sort of uniformity or determinism in their appearances. And let us assume that the best quantitative
statement embodying the concept of organization is
a principle which states that the cost of operation of
physiological systems tends to be a minimum. . . ” Murray illustrated his ideas by postulating a cost function
for “operating an arterial segment.” He proposed that
the radius of a blood vessel results from a compromise
between the advantage of increasing the lumen, which
reduces the resistance to flow and thereby the workload
on the heart, and the disadvantage of increasing overall
blood volume, which increases the metabolic demand of
maintaining the blood (e.g., red blood cells have a life-
span of a few months in humans, which necessitates
a continual production and removal of cells). Murray’s
findings suggest that “. . . the flow of blood past any section shall everywhere bear the same relation to the cube
of the radius of the vessel at that point.” Recently, it has
been shown that Murray’s ideas are consistent with the
observation that the lumen of an artery appears to be
governed, in part, so as to keep the wall shear stress at
a preferred value – for a simple, steady, incompressible,
laminar flow of a Newtonian fluid in a circular tube,
the wall shear stress is proportional to the volumetric
flow rate and inversely proportional to the cube of the
radius [7.19]. Clearly, optimization approaches should
be given increased attention, particularly with regard to
the design of useful biomechanical experiments and the
reduction of the associated data.
Perhaps best known for inventing the Turing machine for computing, Turing also published a seminal
paper on biological growth [7.51]. Briefly, he was interested in mathematically modeling morphogenesis,
that is, the development of the form, or shape, of an
organism. In his words, he sought to understand the
mechanism by which “genes . . . may determine the
anatomical structure of the resulting organism.” Turing recognized the importance of both mechanical and
chemical stimuli in controlling morphogenesis, but he
focused on the chemical aspects, especially the reaction kinetics and diffusion of morphogens, substances
such as growth factors that regulate the development
of form. For example, he postulated linear reaction–
diffusion equations of the form
∂M1
= a (M1 − h) + b (M2 − g) + D1 ∇ 2 M1 ,
∂t
∂M2
= c (M1 − h) + d (M2 − g) + D2 ∇ 2 M2 ,
∂t
(7.15)
where M1 and M2 are concentrations of two morphogens, a, b, c, and d are reaction rates, and D1 and D2
are diffusivities; h and g are equilibrium values of M1
and M2 , and t represents time during morphogenesis. It
was assumed that the local concentration of a particular morphogen tracked the local production or removal
of tissue. Numerical examples revealed that solutions
to such systems of equations could “develop a pattern
or structure due to an instability of the homogeneous
equilibrium, which is triggered off by random disturbances.” These solutions were proposed as possible
descriptors of the morphogenesis. It was not until the
Biological Soft Tissues
1980s, however, that there was an increased interest in
the use of reaction–diffusion models to study biological
growth and remodeling, which now includes studies of
wound healing, tumor growth, angiogenesis, and tissue
engineering in addition to morphogenesis [7.52–56].
As noted by Turing, mechanics clearly plays an important role in such growth and remodeling, thus it is
not surprising that there has been a trend to embed the
reaction–diffusion framework within tissue mechanics
(albeit often within the context of linearized elasticity
or viscoelasticity). For example, Barocas and Tranquillo [7.53] suggested that reaction–diffusion models
for spatial–temporal information on cells could be combined with a mixture theory representation of a tissue
consisting of a fluid constituent and solid network. In
this way, they studied mechanically stimulated cell migration, which is thought to be an early step towards
mechanically stimulated changes in deposition of structural proteins. See the original paper for details.
In summary, there has been significant attention
to modeling the production, diffusion, and half-life of
a host of molecules (growth factors, cytokines, proteases) and how they affect cell migration, mitosis,
apoptosis, and the synthesis and reorientation of extracellular matrix. In some cases the reaction–diffusion
models are used in isolation, but there has been a move
towards combining such relations with those of mechanics (mass and momentum balance). Yet because
of the lack of attention to the finite-strain kinematics
and nonlinear material behavior characteristic of soft
tissues [7.57], there is a pressing need for increased
generalization if this approach is to become truly predictive. Moreover, there is a pressing need for additional
biomechanical experiments throughout the evolution of
the geometry and properties so that appropriate kinetic
equations can be developed.
7.3.2 Kinematic Growth
being conserved, the overall mass density appeared to
remain constant, thus focusing attention on changes in
volume.) Tozeren and Skalak [7.59] suggested further
that finite-strain growth and remodeling in a soft tissue
(idealized as fibrous networks) could be described, in
part, by considering that “The stress-free lengths of the
fibers composing the network are not fixed as in an inert elastic solid, but are assumed to evolve as a result of
growth and stress adaptation. Similarly, the topology of
the fiber network may also evolve under the application
of stress.” One of the remarkable aspects of Skalak’s
work is that he postulated that, if differential growth
is incompatible, then continuity of material may be restored via residual stresses. Residual stresses in arteries
were reported soon thereafter (independently in 1983
by Vaishnav and Fung; see the discussion in [7.19]),
and shown to affect dramatically the computed stress
field in the arterial wall. The basic ideas of incompatible
kinematic growth, residual stress, and evolving material
symmetries and stress-free configurations were seminal
contributions. Rodriguez et al. [7.60] built upon these
ideas and put them into tensorial form – the approach
was called “finite volumetric growth”, which is now
described in brief.
The primary assumption is that one models volumetric growth through a growth tensor Fg , which
describes changes between two fictitious stress-free
configurations: the original body is imagined to be fictitiously cut into small stress-free pieces, each of which
is allowed to grow separately via Fg , with det Fg = 1.
Because these growths need not be compatible, internal forces are often needed to assemble the grown
pieces, via Fa , into a continuous configuration. This,
in general, produces residual stresses, which are now
known to exist in many soft tissues besides arteries.
The formulation is completed by considering elastic
deformations, via Fe , from the intact but residually
stressed traction-free configuration to a current configuration that is induced by external mechanical loads.
The initial–boundary value problem is solved by introducing a constitutive relation for the stress response to
the deformation Fe Fa , which is often assumed to be
isochoric and of the Fung type, plus a relation for the
evolution of the stress-free configuration via Fg (actually Ug since the rotation Rg is assumed to be I). Thus,
growth is assumed to occur in stress-free configurations
and typically not to affect material properties. See, too,
Lubarda and Hoger [7.61], who consider special cases
of transversely isotropic and orthotropic growth.
Among others, Taber [7.62] and Rachev et al. [7.63]
independently embraced the concept of kinematic
179
Part A 7.3
In a seminal paper, Skalak [7.58] offered an approach
very different from the reaction–diffusion approach, one
that brought the analysis of biological growth within
the purview of large-deformation continuum mechanics. He suggested that “any finite growth or change
of form may be regarded as the integrated result of
differential growth, i. e. growth of the infinitesimal elements making up the animal and plant.” His primary
goal, therefore, was to “form a framework within which
growth and deformation may be discussed in regard
to the kinematics involved.” (Note: Although it was
realized that mass may change over time, rather than
7.3 Growth and Remodeling – A New Frontier
180
Part A
Solid Mechanics Topics
growth and solved initial–boundary value problems relating to cardiac development, arterial remodeling in
hypertension and altered flow, and aortic development.
For the purposes of discussion, briefly consider the
model of aortic growth by Taber [7.62]. The aortic
wall was assumed to have material properties (given by
Fung’s exponential relation) that remained constant during growth, which in turn was modeled via additional
constitutive relations for time rates of change of the
growth tensor Fg = diag[λgr , λgθ , 1], namely
dλgr
1
ge t¯θθ (s) − t¯θθ ,
=
dt
Tr
dλgθ
1
ge t¯θθ (s) − t¯θθ
=
dt
Tθ
1
ge +
τ̄w (s) − τ̄w eα(R/Ri −1) ,
Tτ
(7.16)
where Ti are time constants, t¯jj and τ̄w are mean
values of wall stress and flow-induced wall shear
stress, respectively, the superscript ‘ge’ denotes growthequilibrium, α is a parameter that reflects the intensity
of effects, at any undeformed radial location R, of
growth factors produced by the cells that line the arterial wall and interact directly with the blood. Clearly,
growth (i. e., the time rate of change of the stress-free
configuration in multiple directions) continues until the
stresses return to their preferred or equilibrium values. Albeit not in the context of vascular mechanics,
Klisch et al. [7.64] suggested further that the concept
of volumetric growth could be incorporated within the
theory of mixtures (with solid constituents k = 1, . . . , n
and fluid constituent f ) to describe growth in cartilage. The deformation of constituent k was given by
Fk = Fke Fak Fkg , where Fkg was related to a scalar mass
growth function m k via
t
Part A 7.3
det Fkg (t) = exp
m k dτ,
(7.17)
0
and mass balance requires
dk ρ k
+ ρk div vk = ρk m k ,
dt
dfρf
+ ρ f div v f = 0 .
dt
(7.18)
This theory requires evolution equations for Fkg (or similarly, m k ), which the authors suggested could depend on
the stresses, deformations, growth of other constituents,
etc., as well as constitutive relations for the mass growth
function. It is clear that such a function could be related
to the reaction–diffusion framework of Turing, and thus
chemomechanical stimulation of growth. Although it is
reasonable, in principle, to consider a full mixture theory given that so many different constituents contribute
to the overall growth and structural stability of a tissue
or organ, it is very difficult in practice to prescribe appropriate partial traction boundary conditions and very
difficult to identify the requisite constitutive relations
for momentum exchanges. Indeed, it is not clear that
there is a need to model such detail, such as the momentum exchange between different proteins that comprise
the extracellular matrix or between the extracellular matrix and a migrating cell, for example, particularly given
that such migration involves complex chemical reactions (e.g., degradation of proteins at the leading edge of
the cell) not just mechanical interactions. For these and
other reasons, let us now consider an alternative mixture
theory.
7.3.3 Constrained Mixture Approach
Although the theory of kinematic growth yields many
reasonable predictions, we have suggested that it models consequences of growth and remodeling (G&R), not
the processes by which they occur. G&R necessarily
occur in stressed, not fictitious stress-free, configurations, and they occur via the production, removal,
and organization of different constituents; moreover,
G&R need not restore stresses exactly to homeostatic
values. Hence, we introduced a conceptually different
approach to model G&R, one that is based on tracking
the turnover of individual constituents in stressed configurations [7.65]. Here, we illustrate this approach for
2-D (membrane-like) tissues [7.66]. Briefly, let a soft
tissue consist of multiple types of structurally important
constituents, each of which must deform with the overall tissue but may have individual material properties
and associated individual natural (i. e., stress-free) configurations that may evolve over time. We employ the
concept of a constrained mixture wherein constituents
deform together in current configurations and tacitly
assume that they coexist within neighborhoods over
which a local macroscopic homogenization would be
meaningful. Specifically, not only may different constituents coexist at a point of interest, the same type
of constituent produced at different instants can also
coexist. Because of our focus on thin soft tissues consisting primarily of fibrillar collagen, one can consider
a constitutive relation for the principal Cauchy stress resultants for the tissue (i. e., constrained mixture) of the
Biological Soft Tissues
form
T1 (t) = T10 (t) +
1 ∂wk
,
λ2 (t)
∂λ1 (t)
k
1 ∂wk
,
T2 (t) = T20 (t) +
λ1 (t)
∂λ2 (t)
(7.19)
k
where Ti0 (i = 1, 2) represent contributions by an
amorphous matrix (e.g., elastin-dominated or synthetic/reconstituted in a tissue equivalent) that can
degrade but cannot be produced, λi are measurable principal stretches that are experienced by the tissue, and
wk is a stored energy function for collagen family k,
which may be produced or removed over time. Note,
too, that
2 2
(7.20)
λk (t) = λ1 cos α0k + λ2 sin α0k
are stretches experienced by fibers in collagen family
k relative to a common mixture reference configuration, with α0k the angle between fiber family k and
the 1 coordinate axis. To account for the deposition of
new collagen fibers within stressed configurations, however, we further assume the existence of a preferred
(i. e., homeostatic) deposition stretch G kh , whereby
the stretch experienced by fiber family k, relative
to its unique natural configuration, can be shown to
be
λkn(τ) (t) = G kh λk (t)/λk (τ) ,
(7.21)
with t the current time and τ the past time at which
family k was produced. Finally, to account for continual production and removal, let the constituent stored
energies be [7.66]
wk (t) =
0
(7.22)
where ρ is the mixture mass density, M k (0) is the
2-D mass density of constituent k at time 0, when
G&R commences, Q k (t) ∈ [0, 1] is the fraction of
constituent k that was produced before time 0 but
survives to the current time t > 0, m k is the current
mass density production of constituent k, W k λkn(τ) (t)
is the strain energy function for a fiber family relative to its unique natural configuration, and q k is
181
an associated survival function describing that fraction of constituent k that was produced at time τ
(after time 0) and survives to the current time t.
Hence, consistent with (7.4), the principal Cauchy
stress resultants of the constituents that may turnover
are
k
M (0)Q k (t)G kh ∂W k ∂λk (t)
1
T1k (t) =
λ2 (t)
ρλk (0)
∂λkn(τ) (t) ∂λ1 (t)
t
m k (τ)q k (t − τ)G kh
ρλk (τ)
0
∂W k ∂λk (t)
× k
dτ ,
∂λn(τ) (t) ∂λ1 (t)
k
M (0)Q k (t)G kh ∂W k ∂λk (t)
1
k
T2 (t) =
λ1 (t)
ρλk (0)
∂λkn(τ) (t) ∂λ2 (t)
+
t
m k (τ)q k (t − τ)G kh
ρλk (τ)
0
∂W k ∂λk (t)
× k
dτ .
∂λn(τ) (t) ∂λ2 (t)
+
(7.23)
As in most other applications of biomechanics, the
key challenge therefore is to identify specific functional forms for the requisite constitutive relations,
particularly the individual mass density productions,
the survival functions, and the strain energy functions
for the individual fibers, not to mention relations for
muscle contractility and its adaptation. Finally, there
is also a need to prescribe the alignment of newly
produced fibers, not just their rate of production and
removal. These, too, will require contributions from
experimental biomechanics. Illustrative simulations are
found nonetheless in the original paper [7.66], which
show that stable versus unstable growth and remodeling can result, depending on the choice of constitutive
relation.
Given that the biomechanics of growth and remodeling is still in its infancy, it is not yet clear which
approaches will ultimately prove most useful. The interested reader is thus referred to the following as
examples of alternate approaches [7.67–71]. Finally, it
is important to emphasize that, regardless of the specific
theoretical framework, the most pressing need at present
is an experimental program wherein the evolving mechanical properties and geometries of cells, tissues,
and organs are quantified as a function of time during
adaptations (or maladaptations) in response to altered
mechanical loading, and that such information must be
Part A 7.3
M k (0) k
Q (t)W k λkn(0) (t)
ρ
t k
m (τ) k
+
q (t − τ)W k λkn(τ) (t) dτ ,
ρ
7.3 Growth and Remodeling – A New Frontier
182
Part A
Solid Mechanics Topics
correlated with changes in the rates of production and
removal of structurally significant constituents, which
in turn depend on the rates of production, removal, and
diffusion of growth factors, proteases, and related substances. Clearly, biomechanics is not simply mechanics
applied to biology; it is the extension, development, and
application of mechanics to problems in biology and
medicine, which depends on theoretically motivated experimental studies that seek to identify new classes of
constitutive relations.
7.4 Closure
In summary, much has been accomplished in our quest
to quantify the biomechanical behavior of soft tissues, yet much remains to be learned. Fortunately,
continuing technological developments necessary for
advancing experimental biomechanics (e.g., optical
tweezers, atomic force and multiphoton microscopes,
tissue bioreactors) combined with traditional methods
of testing (e.g., computer-controlled in-plane biaxial
testing of planar specimens, inflation and extension
testing of tubular specimens, and inflation testing of
membranous specimens; see [7.19] Chap. 5) as well
as continuing advances in theoretical and computational mechanics are helping us to probe deeper into
the mechanobiology and biomechanics every day. Thus,
both the potential and the promise of engineering contributions have never been greater. It is hoped, therefore,
that this chapter provided some background, and especially some motivation, to contribute to this important
field. The interested reader is also referred to a number
of related books, listed in the Bibliography, and encouraged to consult archival papers that can be found
in many journals, including Biomechanics and Modeling in Mechanobiology, the Journal of Biomechanics,
and the Journal of Biomechanical Engineering. Indeed,
an excellent electronic search engine is NIH PubMed,
which can be found via the National Institutes of Health
web site (www.nih.gov); it will serve us well as we
continue to build on past achievements.
7.5 Further Reading
Part A 7.5
Given the depth and breadth of the knowledge base
in biomedical research, no one person can begin to
gain all of the needed expertise. Hence, biomechanical
research requires teams consisting of experts in mechanics (theoretical, experimental, and computational)
as well as biology, physiology, pathology, and clinical practice. Nevertheless, bioengineers must have a
basic understanding of the biological concepts. I recommend, therefore, that the serious bioengineer keep
nearby books on (i) molecular and cell biology, (ii) histology, and (iii) medical definitions. Below, I list some
books that will serve the reader well.
•
•
•
•
H. Abe, K. Hayashi, M. Sato: Data Book on Mechanical Properties of Living Cells, Tissues, and
Organs (Springer, New York 1996)
Dorland’s Illustrated Medical Dictionary (Saunders, Philadelphia 1988)
S.C. Cowin, J.D. Humphrey: Cardiovascular Soft
Tissue Mechanics (Kluwer Academic, Dordrecht
2001)
S.C. Cowin, S.B. Doty: Tissue Mechanics (Springer,
New York 2007)
•
•
•
•
•
•
•
•
•
D. Fawcett: A Textbook of Histology (Saunders,
Philadelphia 1986)
Y.C. Fung: Biomechanics: Mechanical Properties of
Living Tissues (Springer, New York 1993)
F. Guilak, D.L. Butler, S.A. Goldstein, D.J. Mooney:
Functional Tissue Engineering (Springer, New York
2003)
G.A. Holzapfel, R.W. Ogden: Biomechanics of Soft
Tissue in Cardiovascular Systems (Springer, Vienna
2003)
G.A. Holzapfel, R.W. Ogden: Mechanics of
Biological Tissue (Springer, Berlin, Heidelberg
2006)
J.D. Humphrey, S.L. Delange: An Introduction to
Biomechanics (Springer, New York 2004)
V.C. Mow, R.M. Hochmuth, F. Guilak, R. TransSon-Tay: Cell Mechanics and Cellular Engineering
(Springer, New York 1994)
W.M. Saltzman: Tissue Engineering (Oxford Univ
Press, Oxford 2004)
L.A. Taber: Nonlinear Theory of Elasticity: Applications to Biomechanics (World Scientific, Singapore 2004)
Biological Soft Tissues
References
183
References
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.16
7.17
7.18
7.19
7.20
7.21
7.22
7.23
7.24
7.25
7.26
7.27
7.28
7.29
7.30
7.31
7.32
7.33
7.34
7.35
J.R. Walton, J.P. Wilber: Sufficient conditions for
strong ellipticity for a class of anisotropic materials,
Int. J. Nonlinear Mech. 38, 441–455 (2003)
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