AMER. ZOOL., 24:13-22 (1984)
Structure and Stress-Strain Relationship of Soft Tissues 1
Y. C. FUNG
Department ofAMES /Bioengtneenng, University of California, San Diego,
La folia, California 92093
SYNOPSIS. The mechanical properties of a soft tissue are related to its structure. We shall
illustrate this by the properties of the arteries and the lung. Viscoelasticity, strain rate
effects, pseudo-elasticity, and constitutive equations are discussed. The mecahnical properties of an organ is, however, not only based on the tissues of the organ, but also on its
geometry and relationship to the neighboring organs. A typical example is the blood
vessel. The capillary blood vessels of the mesentery are "rigid"; those in the bat's wing
are "distensible"; whereas the capillaries of the lung are "sheet" like: rigid in one plane,
and compliant in another. The stress-strain relationship of the systemic arteries is highly
nonlinear, stiffening exponentially with increasing strains; yet that of the pulmonary
arteries in the lung is linear. The systemic veins are easily collapsible; yet the pulmonary
veins in the lung are not: they remain patent when the blood pressure falls below the
alveolar gas pressure. The explanation of these differences lies in the varied interactions
between the blood vessels and the surrounding tissues in different organs. The implications
of these differences on blood circulation are pointed out. The role of ultrastructure is
discussed.
INTRODUCTION
strain relationship of the lung tissue of the
dog (with the airspace filled with saline so
that the surface tension between the alveolar gas and the moist alveolar walls is
replaced by the very small liquid-solid
interfacial tension. The tissue was prepared in the form of a slab, and biaxial
loading was used, while the strains were
monitored in the middle portion of the
specimen, away from the edges (in order
to avoid the "edge effect" as much as possible). After a number of cycles of loading
and unloading, a repeatable stress-strain
loop as shown in Figure 1 was obtained.
The existence of the loop shows that the
tissue is viscoelastic, and not elastic. But
since the loop is repeatable we can treat
the loading and unloading curves separately and borrow the method of the theory of elasticity to describe the mechanical
Some general features
Some features of the mechanical prop- properties. Hence the term "pseudoelaserties are common to all soft tissues. They ticity." Figure 2 shows the stress-strain
are pseudo-elastic, that is, they are not elas- relationship of the same lung tissue in loadtic, but under periodic loading and unload- ing at different strain rates. Each cycle was
ing a steady-state stress-strain relationship done at a constant rate. The period of each
exists which is not very sensitive to strain cycle is noted in the figure. It is seen that
over a 360-fold change in strain rate there
rate. For example, Figure 1, from Vawter, was only a minor change in the stress-strain
Fung and West (1978), shows the stress- relationship. The hysteresis, H, defined as
the ratio of the area of the hysteresis loop
divided
by the area under the loading curve,
' From the Symposium on Biomechanics presented
at the Annual Meeting of the American Society of is also noted in Figure 2. H is seen to be
Zoologists, 27-30 December 1982, at Louisville, Ken- variable, but its variation with strain rate
Soft tissues are major components of animal body: the muscle makes locomotion
possible. The skin protects the internal
milieu. A variety of soft tissues make up
the internal organs. The function of all
organs is closely related to the mechanics
of soft tissues, about which this article is
concerned.
Soft tissues are made of collagen, elastin,
muscle and other cells, and ground substances. Their mechanical properties depend not only on their chemical composition, but also on structural details. For
organs, their mechanical property depends
not only on their own materials and structures, but also on the environment. We
shall illustrate this with several examples.
tucky.
13
14
Y. C. FUNG
CYCLE TIME.
60 -i
HYSTERESIS
STRETCH
8 0 -1 •
RELEASE
18 SEC, M = 02T
60 SEC, H = 0 J2
x 40 -
220 SEC.H . 030
«
60 -
900 SEC, H = 028
6500 SEC, H = O35
' *
S 40 -
it
5
08
10
12
14
X , (EXTENSION RATIO L , / L 0 1 ) ,
16
18
OIMENSIONLESS
FIG. 1. A typical stress-strain curve for uniaxial loading. Every fourth data point is plotted. Note that the
unloading curve is different from the loading curve,
showing the existence of hysteresis. From Vawter el
al. (1978), by permission.
is not large. Similar experience is encountered with other tissues. Records of skeletal and cardiac muscles, ureter, taenia coli,
arteries, veins, pericardium, mesentery, bile
duct, skin, tendon, elastin (lig. nuchae with
collagen denatured), cartilage, and other
tissues show the same characteristics. The
stress-strain relationships of some tissues
have been tested in a range of strain rate
covering a million-fold difference between
the slowest and the fastest cycling, and the
stresses at the same strain are usually found
to differ by less than a factor of 2. The
fastest stress cycle can be imposed by ultrasound, and it is known that for most tissues
the damping per cycle of oscillation remains
almost constant as frequency varies. The
slowest cycling in the laboratory is often
done by step-by-step testing with a long
period of waiting between steps. Hysteresis
loops do not vanish in these "static" tests;
in fact, they usually remain comparable in
size to those obtained at moderate frequencies.
The features shown in Figures 1 and 2
may be described by saying that living soft
tissues are nonlinearly pseudo-elastic. The
stress-strain relationship is nonlinear, the
viscoelasticity is pseudoelastic—hysteresis
may be sizable, but it varies only mildly
over a wide range of strain rates.
20 •
08
10
II
12
14
X, (STRETCH RATIO l , / l
16
0
FIG. 2. Loading phase at different strain rates. Varying strain rate over 2.5 decades caused only small
changes in response. The hysteresis, H, is the ratio
of the area of hysteresis loop (not shown) to the area
under the loading curve. The period of cycling and
the values of H are given in the insert. From Vawter
et al. (1978), by permission.
method of elasticity to describe the stressstrain relationship. For a nonlinear material the simplest way is to introduce a
pseudo-elastic potential (also called a strain
energy function), p0W, which is a function
of the Green's strain components E,r The
partial derivatives of p0W with respect to
E,j gives the corresponding stresses Su
(Kirchoff stresses). W is denned for a unit
mass of the tissue, p0 is the density of the
tissue in the initial state, hence p0W is the
strain energy per unit initial volume. Thus
(i,j = l , 2 , 3).
(1)
If the material is incompressible (volume
does not change) then k can take on a pressure that is independent of the deformation
of the body. In that case a pressure term
should be added to the right hand side of
Equation (1). The value of the pressure (as
in water) can vary from point to point, and
it can be determined from the equations
of motion and continuity, and boundary
conditions.
Xonlinear elasticity
An example of pseudo-elastic potential
Treating the loading and unloading for arteries and veins is the following (Fung
curves separately, we can borrow the etal, 1979):
SOFT TISSUE MECHANICS
p0W<2> = C exp[a,E,2 + a2E22 + 2a4E,E2]
(2)
<2)
Here the superscript (2) over p0W signifies that this is a two-dimensional approximation, which can yield only a relationship
between the average circumferential and
axial stresses (S,, S2) and strains (E,, E2).
Differentiation of p0W<2) with respect to E,
yields S,, that with respect to E2 yields S2.
Figure 3 shows a comparison of the fitting
of Equation (1) to experimental data on
rabbit arteries subject to increasing internal pressure and longitudinal stretching.
The constants C, a,, a2, a4 are the material
constants that characterize the artery.
Other forms of strain energy function
such as polynomials can be used which can
also yield good fitting with experimental
results. Most soft tissues can be described
by a strain energy function similar to Equation (1).
For a body subjected to small changes in
strains, the corresponding changes in
stresses are also small and the relation
between the incremental stresses and
strains can be linearized if the strains are
sufficiently small. The linearized relationship is the Hooke's law, for which the
familiar material constants are the incremental Young's modulus and incremental
shear modulus. For soft tissues a general
feature implied by Equation (2) is that the
incremental moduli increase with increasing stresses.
Viscoelasticity
We have shown in Figures 1 and 2 that
soft tissues are viscoelastic in a special way:
the hysteresis loop is relatively insensitive
to strain rate. If relaxation under constant
strain is measured it will be found that elastin relaxes very little, tendon, mesentery,
skin, blood vessels, lung, and smooth muscles relax more and more in the order listed.
Creep under constant load exists for all
these tissues.
A mathematical model of viscoelasticity
of a tissue must cover all features of hysteresis, relaxation, and creep. One of the
most popular models of viscoelasticity is
the Maxwell model of a spring in series
with a dashpot. The other is the Voigt
15
model with a spring and a dashpot in parallel. A third is the Kelvin model which is
a combination of a spring in parallel with
a Maxwell body (see Fig. 4a, b, c). None of
these can represent a soft tissue, because
when a material represented by any one of
these models is subjected to a cyclic strain,
the hysteresis will not be insensitive to strain
rate: as frequency increases the dashpot in
the Maxwell body will move less and less
at same load so the hysteresis decreases with
frequency. On the other hand, the Voigt
body will let the dashpot take up more and
more of of the load so that the hysteresis
increases with frequency (see Fig. 4d, e).
For the Kelvin body there exists a characteristic frequency at which the hysteresis
is a maximum (see Fig. 4f). None of these
has the feature of nearly constant hysteresis as soft tissues do.
A model suitable for the soft tissue is
shown in Figure 4g, which has an infinite
number of springs and dashpots. In the
corresponding hysteresis diagram shown
in Figure 4h there are an infinite number
of bell-shaped curves which add up to a
continuous curve of nearly constant height
over a very wide range of frequencies. In
this situation, we say that the soft tissue has
a continuous relaxation spectrum. The two
ends of the spectrum, marked by frequencies T," 1 and T 2 ~' in Figure 4h, define two
characteristic times T, and T2 which can be
determined from experimental data (see
Fung [1972; 1981, pp. 232 et. seq.} for
mathematical details). Tanaka and Fung
(1974) have found T, and T2 for various
arteries of the dog: T, lies in the range of
several hundred to thousands of seconds,
T2 lies in the range of 0.05 to 0.36 sec. Chen
and Fung (1973) showed that for the mesentery T], T2 are 1.869 x 104 sec and
1.735 x 10~5 sec. Woo et al. (1979) have
found that for the cartilage T2 = O.006 sec,
r, = 8.38 sec.
Why different blood vessels
behave so differently
One of the beauties and puzzles of the
biological world is that it has a great variety
of things, and sometimes, the same thing
has quite different properties in different
circumstances. Take blood vessels as an
16
Y. C. FUNG
a
carotid
a'
o
left iliac
d
A
lower aorta
^
«
upper aorta
^
Experimental
Theoretical (exponential)
100
^
75
CO
50
25
.0
0.2
0.4
Green's Strain, EQQ
0.6
FIG. 3. Comparison of the stress-strain relationships obtained from Eqs. (2) and (1) with experimental data
on four normal rabbit arteries. The symbols are defined in the inset. The curves joining experimental data
points are not always smooth because stresses and strains in two dimensions are coupled and any disturbance
in EK results in a kink in the SM vs. EM curve, and vice versa. From Fung et al. (1979), by permission.
example. The aorta and thoracic arteries illary sheet with the transmural pressure
have nonlinear stress-strain curves as shown AP (blood pressure minus alveolar gas presin Figure 3. The pulmonary arteries and sure) is shown. The pulmonary capillaries
veins, in contrast, have linear pressure- are closely knit into a dense network which
diameter relationships as shown in Figure occupies about 90% of the total space in
5, though this does not imply a linear stress- the interalveolar septa (the alveolar walls).
strain curve. The capillary blood vessels of It is best to describe this network as a sheet.
the mesentery appears to be rigid—with- Each sheet is exposed to gas on both sides.
out measurable change in diameter when The sheet thickness varies with the transblood pressure changes over a range of 100 mural pressure. Figure 6 shows that the
mm Hg (Baez, 1960; Fung et al, 1966). thickness of the pulmonary capillary sheets
But the capillary blood vessels in the lung of the cat increases linearly with increasing
are very distensible (see Fig. 6), in which transmural pressure at a rate of 0.22 ^m
the variation of the thickness of the cap- per cm H 2 O when AP is positive. But when
17
SOFT TISSUE MECHANICS
n
carotid
eC
o
left iliac
<nt
A
lower aorta
if
«
upper aorta
<f
Experimental
Theoretical (exponential)
50
40
30
V)
V)
£
CO
20
10
0.5
0.7
0.9
Green's Strain, E n
FIG. 3.
1.1
Continued.
AP is negative the thickness quickly drops
to zero. There is a sudden change of thickness at AP = 0. When AP tends to 0 from
the positive side the limiting value of the
sheet thickness is 4.28 Mm for the cat. When
AP < — 1 cm H2O the capillaries are all
collapsed. Why do the capillaries of the
lung behave so differently from those of
the peripheral circulation?
There is another important property of
the blood vessels that has an important
physiological effect: the stability of the vessel when the external pressure exceeds the
internal pressure. We have seen in Figure
6 that the pulmonary capillaries collapse
when AP < 0. But we know that peripheral
capillaries do not collapse when blood pressure falls below tissue pressure (see Baez,
1964; Fung et al., 1966). On the other hand,
it is common knowledge that peripheral
veins and vena cava collapse when the blood
pressure falls below the pressure in the surrounding media. But the pulmonary veins
do not collapse when the airway pressure
exceeds the blood pressure, as the data
shown in Figure 5 demonstrates. We have
shown further (Fung et al, 1982) that pulmonary venules do not collapse under the
same condition. Why do these vessels
behave so differently while their composition and anatomical and histological
structures are very similar?
18
Y. C. FUNG
fcn~
which provides little resistance to deformation; and as a consequence the pulmonary capillaries are distensible and collapsible (Fung and Sobin, 19726).
If we compare a pulmonary artery with
a peripheral artery, we see that the latter
is an isolated vessel, whereas the pulmoInf
nary vessel is embedded in the lung parenInf
chyma. The lung tissue (alveolar walls) pulls
on the pulmonary blood vessel wall, and
keeps it patent when the alveolar gas pressure exceeds the blood pressure; and the
tissue stress influences the vessel deformation so much that a nearly linear
pressure-diameter relationship (Fig. 5) is
The In + I llh
obtained.
If the pulmonary artery were isomember
lated,
it
would
most likely behave the same
(S)
way as the peripheral vessels (Fig. 3).
The same reason explains the difference
between the incollapsibility of pulmonary
veins versus the collapsibility of peripheral
veins: the pulmonary veins have the support of the lung parenchyma while the
FIG. 4. Several models of viscoelasticity and their peripheral veins have little support from
corresponding hysteresis-frequency relationships, (a) their neighbors.
Maxwell model, (b) Voight model, (c), Kelvin model,
(d), (e), (f) hysteresis of models a, b, c respectively.
The hysteresis (H) is defined as the ratio of the area
of hysteresis loop to the area under the loading curve.
The abscissa is frequency in log scale, (g) is a generalized model for living soft tissues, (h) is the corresponding hysteresis vs. frequency curve. Each
member of the generalized model contributes a small
bell-shaped curve. The sum of these small bell-shaped
curves yields a curve of H which is almost constant
over a wide range of frequencies. See text for explanation.
Physiological importance of
vessel elasticity
It may not be out of place to record a
couple of useful but not too well-known
formulas here to show the importance of
the distensibility of blood vessel on blood
flow. If the radius of a cylindrical blood
vessel, a, varies linearly with the transmural pressure, AP, in such a way that
a = a0 + aAP
Influence of neighbors
The answer to the questions named
above is that the property of a blood vessel
depends not only on the intrinsic properties of the blood vessel wall, but also on the
properties of neighboring tissues. The
peripheral capillaries in the mesentery are
embedded in a gel. A gel behaves as a solid;
and, as a consequence, a capillary blood
vessel embedded in it behaves as a tunnel
in solid earth. That is why the capillary is
so rigid with respect to blood pressure, and
so stable with respect to compression (Fung
et al., 1966). On the other hand, a pulmonary capillary is exposed to alveolar gas
(3)
where a is the compliance constant, then
theflowin the vessel is given by Fung (1982)
to be
Q =
[(a0 + aApentry)5
- (a0 + aApeM)5]
(4)
where a0 is the radius when Ap = 0, Ap =
blood pressure minus external pressure,
L = length of the blood vessel, n is the viscosity of the blood, and the subscripts
"entry" and "exit" refer to the entry and
exit sections of the vessel. The power 5 is
a powerful factor. If the compliance is large,
the effect of vessel elasticity on flow can be
very large!
19
SOFT TISSUE MECHANICS
rt
= -5 or HfO fSl Pal
• 100200**
• 200 400 um
1 l5 °
100
, - P 4 cm H,0 198 Pal
> cm H,0 (98 Pa)
-15
-10
200
Pn = -20 cm H,0 C 98 Pal
• 100200 um
A 200400iim
, = - ; f lOTH2O I* 38 Pa)
• 100 200 um
• 300-400 ttm
800t200nm
HIM t SO
I
I
100
0
= P, - P» cm H,0 (98 Pal
10
AP = Pv - P t cm H,0 |98 Pa)
FIG. 5. The extensibility of pulmonary veins subjected to positive and negative pv — pA. The vessel diameter
is normalized against its value when pv - pPL is 10 cm H2O at which the vessel cross section is circular. The
points are classified according to their diameters at pv — pPL = 1 0 . From Yen and Fopiano (1981), by permission.
For the pulmonary capillaries the blood
vessels are arranged in two-dimensional
sheets, and the total flow is given by the
formula (Fung and Sobin, 1972a):
area
- (h0 + aApexil)*].
(5)
Here h denotes the alveolar sheet thickness, which, as is shown in Figure 6, varies
linearly with Ap when Ap > 0:
(6)
h = h0 + aAp.
a is the compliance constant of the alveolar
capillaries. Other symbols in Equation (5)
are: Area = area of the alveolar wall, \i is
viscosity of blood, k and f are numerical
factors which depend on the sheet geom-
etry, L"2 is the mean path length of blood
between the entry and exit sections. In the
lung, the entry section is located at the
pulmonary arterioles, the exit is located at
the pulmonary venules.
Equation (5), together with the knowledge that pulmonary veins and venules do
not collapse when the pulmonary alveolar
gas pressure exceeds the blood pressure,
explains an important phenomenon which
is known by the name of "waterfall" or
"sluicing." Figure 7 illustrates the phenomenon: Let a lung be perfused with fixed
airway pressure (pA) and arterial pressure
(pa), while the left atrium pressure pv is
varied. Let the flow be measured when pv
is gradually reduced. When pv > pa there
is no flow. When pv is decreased below p2
flow starts, and it increases with decreasing
20
Y. C. FUNG
HICt(NESS
s l2
=J-II
i—
i—
LU
LLJ
CO
10
9
8
7
6
h = 4 28+0 219 Ap (CAT)
•MEAN±°STD DEV
5
Approx used in this paper
h~=h0 +aAp when Ap>0
~h~z 0 when Ap*s- 0
=- o -5
I
5
I
I—I
I
I
10
I
I
I
I
I
I
15
I
I
I
I
I
20
I
I
l_l
25
I
I
I
I
I
I
1 1—I
30
L_l
35
1 I
I
I
I
L-
40
Ap,CAPILLARY-ALVEOLAR PRESSURE, c m H 2 0
FIG. 6. The variation of the thickness of the pulmonary capillary sheet of the cat with the transmural pressure
Ap = capillary blood pressure minus the alveolar gas pressure. From Fung and Sobin (1972), by permission.
pv. But an upper limit is reached when pv
becomes equal to pA. From there on further decrease of pv does not increase the
flow: a flow limitation is reached. This is
analogous to a waterfall whose volume flow
rate does not depend on the height of the
fall. The explanation lies in Equation (5):
when pv < p a , Ah vanishes and the flow
Fie. 7. The variation of the blood flow in the lung,
Q, with decreasing left atrium pressure (pvp) at three
fixed values of pulmonary arterial pressures. As pvp
decreases, the flow reaches a plateau, resulting in a
phenomenon called "vascular waterfall." From Permutt et al. (1962), by permission.
depends entirely on Ap at the pulmonary
arteriole. For further decrease in pv the
pulmonary veins and venules do not collapse, only the capillaries can collapse. The
site of flow limitation, or sluicing gate, must
be located at the junctions of capillaries
and venules. The last term in Equation (5)
is either negligible compared with rest, or
is zero (see Fig. 6), and Q remains constant.
The similarity of Equations (4) and (5)
suggests that other flow limitation phenomena can be similarly explained. This
includes the phenomena of maximum flow
limitation in the airway in forced expiration, and flow limitation in micturition due
to muscle sphincter action in male (with
one sluicing section) and female (with two
sluicing sections) urethra.
Ultrastrudure
All the properties of the organs and tissues have an ultrastructural basis. For soft
tissues, the mechanical properties can be
analyzed ultimately in terms of the networks of collagen and elastin fibers, the
muscle and other cells, and the ground substances and fluids. If the ultrastructure is
known, we should be able to theoretically
21
SOFT TISSUE MECHANICS
deduce the mechanical properties of the
tissue. We have shown above, however, that
at a higher level, from tissues to organs,
we must consider the geometric structure
and interaction of various components. By
analogy, at a lower level, from fibers, cells
and ground substances to tissues, we also
have to consider the geometric structure
and interaction of these components.
Studies of collagen and elastin networks
in arteries were initiated by Roach and
Burton (1957), with a method of differential digestion with enzymes. Later, the
contribution of vascular smooth muscles
was evaluated by means of various vasoactive drugs. But progress has been slow
because structural data about the collagen
and elastin fibers in the tissue are difficult
to obtain. These fibers are densely packed.
Looking at an optical or electron microscopic picture of a selectively digested
artery is like looking at a dense forest in a
landscape. It is difficult to make meaningful measurements. For this reason, our
more recent work has been concentrated
on those tissues in which the collagen and
elastin networks are less dense. Sobin et al.
(1982) and Wall et al. (1981) have systematically photographed these fibers in the
alveolar walls of human lung and obtained
statistical data on fiber width and curvature. These data can then form the basis
for a theory connecting the fine structure
with mechanical properties.
Perspectives
The most crucial step in the development of biomechanics is the identification
of the constitutive equations of the tissues
involved, that is, a concise mathematical
description of the mechanical properties.
If the constitutive equations are known,
then biomechanics problems can be formulated as mathematical problems and
solutions can be definitive. Without constitutive equations, biomechanics will
remain qualitative in character. After the
form of the constitutive equation is determined, the next step is to systematically
collect data on the material constants of
various times. Until we have a complete set
of data on material constants, the power
of biomechanics to predict the function of
an animal will be limited. Only when the
full power of biomechanics to predict the
behavior of an animal when certain parameters are changed is developed can biomechanics be of real service to medicine,
surgery, health preservation and improvement, sports, welfare, and quality oflife of
man and animals.
In this article we have attempted to show
how soft tissues behave and how constitutive equations can be arrived at. We have
used the blood vessels to show that even
though all vessels are fundamentally similar, the constitutive equations and material
constants of one vessel can be very different from that of another because of structural differences and because of interaction with neighboring tissues or organs.
In the immediate future we should complete the program of identifying constitutive equations and material constants, and
develop biomechanics to the point of
becoming a practical tool for the service
of man.
ACKNOWLEDGMENTS
Support of NIH through Grants HL26647 and HL-07089 and NSF through
Grant CME 79-10560 is gratefully
acknowledged.
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22
Y. C. FUNG
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