Indentation Versus Tensile Measurements
of Young’s Modulus for Soft Biological Tissues
Clayton T. McKee, Ph.D.,1Julie A. Last, Ph.D.,2Paul Russell, Ph.D.,1and Christopher J. Murphy, D.V.M., Ph.D.1,3
In this review, we compare the reported values of Young’s modulus (YM) obtained from indentation and tensile
deformations of soft biological tissues. When the method of deformation is ignored, YM values for any given
tissue typically span several orders of magnitude. If the method of deformation is considered, then a consistent
and less ambiguous result emerges. On average, YM values for soft tissues are consistently lower when obtained
by indentation deformations. We discuss the implications and potential impact of this finding.
on the skeletal frame. These soft tissues range from large
organs to small connective tissues with cells and extracel-
lular matrices. They are continually stressed by a multitude
of macro-, micro-, and nanoscopic forces, both internally
and externally, and must be resilient enough to deform
reversibly without damage and still maintain function. For
decades it has been understood that changes in the mac-
roscopic stiffness of tissues can indicate internal disease or
injury. We now understand that changes in the bulk com-
pliance of soft tissues can indicate the onset of conditions
such as breast cancer,1–3atherosclerosis,4–11fibrosis,12or
glaucoma13at a macroscopic level. At the micro- to nano-
scopic level, bulk and local compliance influence a wide
menu of fundamental cell behaviors, including cell mor-
tiation,18,26–29and response to therapeutic agents.30It is
therefore important to properly characterize the compliant
biophysical state of tissues to understand how this bio-
physical attribute relates to proper function. However, a
cursory review of the literature quickly reveals significant
differences in reported values of the compliance of soft
tissues. These variations can influence the understanding of
tissue function and/or failure, interpretation of cellular re-
sponses to biophysical stimuli, or the rational design and
use of biological simulants.31–35The aim of this review is to
determine the origins of these variations. We do not explore
every confounding variable and material property for a
given tissue; rather, the focus of this review is to delineate
the extent to which the experimental method for measuring
modulus affects the interpretation of a commonly quanti-
he rigid skeletal system of vertebrates provides sup-
port and protection for soft tissues that reside within and
fied compliant descriptor, Young’s modulus (YM), by spe-
cifically comparing probe indentation to tensile stretching
measurements of soft biological tissues.
YM describes the ability of an elastic material to resist
deformation to an applied stress. Unfortunately, reported
values of YM for a given tissue can span several orders of
magnitude. The human cornea is a good example, with re-
ported modulus values ranging from 2.9kPa36to 19MPa37
when measured by atomic force microscopy (AFM)38tensile
ing,36,37,44-47This wide variation in reported YM values is not
limited to the cornea. Part of the variation in reported YM
values stems from variation in controllable experimental
variables. Examples include in vivo versus ex vivo measure-
ments, tissue hydration state, time from death/tissue exci-
sion, temperature, storage medium, and the experimental
method used. These experimental differences make direct
comparison of results between studies difficult. Direct com-
parisons are also complicated by the various material prop-
erties that can be used to describe compliance. Again, the eye
is an excellent example; intermixed with reports on intrinsic
material properties such as bulk, shear, or YM are reports on
properties such as ocular or scleral rigidity.48–50Reliance on
empirical values, such as ocular rigidity, can potentially
complicate understanding and diagnosis of disease and
should therefore be used with caution.
Variations in reported YM values for soft tissues may also
stem from the application of elastic models to describe vis-
coelastic responses. YM is commonly used to try and quan-
biomaterials. Strictly speaking, however, YM quantifies the
response of a perfectly elastic material, limiting its use to
metals and crystalline solids or to materials that possess
significant regions of linear stress–strain behavior. These
1Department of Surgical and Radiological Science, School of Veterinary Medicine, University of California Davis, Davis, California.
2Department of Chemical and Biological Engineering, University of Wisconsin–Madison, Madison, Wisconsin.
3Department of Ophthalmology and Vision Science, School of Medicine, University of California Davis, Davis, California.
TISSUE ENGINEERING: Part B
Volume 17, Number 3, 2011
ª Mary Ann Liebert, Inc.
complications have been discussed in detail51and contribute
to confusion and variability on reported mechanical prop-
erties of soft tissues. As noted by Fung,51YM values obtained
by tensile measurements must be accompanied by a state-
ment of the levels of stress and strain applied to the tissue to
be of any quantitative value. The nonlinear response of soft
tissues and the inherent difficulty in obtaining YM values by
tensile deformation has also resulted in the formulation of
nonlinear stress–strain models39,51,52that do not attempt to
define YM as the quantitative descriptive parameter that it is.
However, given the aim of this review, which attempts to
address the effect of experimental method on reported values
of elastic modulus, its inclusion in this review is warranted.
For perfectly elastic materials, YM is defined as the ratio of
applied stress to resultant strain and has units of measure-
ment in N/m2.
Stress is the force divided by the area over which it is
applied. Strain is a dimensionless quantity defined by the
stress-induced change in length of a material divided by its
unstressed length (DL/L). Soft tissues are not perfectly elastic
materials or homogeneous, and they typically display both
viscous and elastic properties that are dependent on time
and typically display nonlinear stress–strain functions (nev-
ertheless, nonlinear functions do not necessarily define a
material as viscoelastic). For perfectly elastic materials a
single YM value defines the response of material to defor-
mation. For soft biological tissues, the resistance to defor-
mation typically increases as the applied stress increases.
Therefore, YM, defined by Equation 1, is not constant and
depends on the specific applied stress, which is particularly
important as tissues in vivo typically exist in a prestressed
state.53As the solution to Equation 1 is dependent on the
applied stress, multiple YM values could be obtained for soft
biological tissues. To avert this problem, soft biological tis-
sues are typically assumed to behave as elastic solids if a
significant linear regime of stress-to-strain exists in the limit
of small strain response to applied stress.
Two methods are commonly used to deform soft tissues:
probe indentation and tensile stretching. Both methods are
employed to describe the compliant response of a material to
an applied stress. They are, however, distinct. Indentation
deformations are maximized at the point of indenter contact
and radially diminish to zero with increasing distance from
the probe, whereas tensile-induced strains span the bulk of
the sample being stressed. If a tissue is not homogeneous
from nanoscopic to macroscopic length scales, the measured
compliance may depend on the experimental technique
employed. This review summarizes previously published
results and documents that differences in methods used are
major contributors to the wide range of values reported for
soft biological tissues. The results highlight that knowledge
of the method used to measure YM is essential for correct
interpretation of the data.
In this review, only those publications that report elastic
modulus values are presented. The nonelastic properties of
these viscoelastic tissues are not included here, as these de-
scriptions of tissue mechanics would make the stated pur-
poseofthis reviewimpossible. However, thetime
dependent, viscous properties of biological tissues are im-
portant and we direct readers to the following reviews and
articles, which discuss these properties for many of the tis-
sues reviewed here.54–62In addition, values, such as the
tangent modulus, obtained from regions of stress–strain
curves that are outside of the elastic regime are not included
in this review. Where possible, we compare YM values ob-
tained when both values (by indentation and tensile
stretching) for a given tissue have been reported (not all
tissues we reviewed have reported values for both methods).
There are a variety of instruments used to indent sam-
ples,63–66ranging from AFMs, capable of applying pico-
newton loads,to nanoindentors,
nanonewton loads, to larger industrial indenters capable of
applying micro to meganewton loads. No single indenter is
ideally suited for every tissue type and ultimately the specific
research objective will dictate which instrument is deemed
appropriate. However, accurate determination of YM re-
quires the specific apparatus be capable of sensitive detection
of the initial point of contact between the indenter and the
sample. The instrument must also have high resolution in the
subsequent changes in load and indentation depth and fine
control of the indentation velocity.
Under the assumption of elastic deformation, YM of a
given biological material is typically determined by fitting
the measured indentation depth as a function of indenter
load during approach.67–70YM values can also be obtained
from the interpretation of unloading curves when the in-
denter is being withdrawn from the sample.71The models
used to fit these indentation curves are indenter geometry
specific; therefore, indenters are objects that are, or can be,
approximated as spheres, cones, or flat cylinders, as the
contact mechanics for these geometries are well estab-
lished.72–78Table 1 lists the elastic model solutions for the
common geometries used to determine YM. A full descrip-
tion of the elastic and viscous properties of a tissue would
require additional measurement of its frequency-dependent
response.79It is instructive to generalize the equations listed
in Table 1 to the following equation:
where F is the force applied by the indenter, d is the inden-
tation depth, a and m are constants where the geometry of
the indenter determines the value of m. The value of m is 1
for flat cylinders, 1.5 for spheres, and 2 for cones. Equation 2
is easily linearized (log–log plot), which allows for a quick
and easy check to ensure that experimental data fits the
correct power law for the indenter geometry used.
Using force versus indentation curves to determine YM
can be complicated due to the viscoelastic nature of a given
biological sample. It is at times difficult to determine the
correct indentation depth over which the sample behaves as
an elastic solid. This region must be defined so that YM can
be accurately determined. For a perfectly elastic material, no
energy is lost to the sample during indentation and both the
loading and unloading curves will be coincident. The elastic
regime of a viscoelastic material can therefore be experi-
mentally determined by controlling the indentation velocity
and depth to produce loading and unloading curves that fall
156McKEE ET AL.
on one another. This result is not always possible as adhe-
sions between the probe and sample can occur that exceed
either the load capacity or drive range of a given apparatus.
This is particularly problematic when using AFMs, which
use very weak springs and piezoelectric crystals with small
drive ranges to move the indenter into a sample. A more
quantitative solution to determine the elastic regime of a
viscoelastic material can be obtained by noting that the
models in Table 1 predict a constant value of E for any in-
dentation depth. Therefore, the ratio of experimental values
of force and indentation can be used to determine the range
over which a specific model applies. For example, Figure 1A
is a plot of indentation force versus depth on a polyacryl-
amide hydrogel prepared in our laboratory and indented
using an AFM cantilever with an incorporated square pyr-
amid tip. Figure 1B is a plot ofpF(1?v2)
2tan(a)d2¼E versus inden-
tation depth, where F is the force, m is Poisson’s ratio, a is the
half angle opening of the AFM tip and d is the depth of
penetration. In using this equation, we have assumed the
square pyramid is a cone. This plot shows that the hydrogel
behaved as an elastic solid with a constant E to *80nm of
indentation. Beyond 80nm, E is no longer constant, indicat-
ing the material is no longer behaving as an elastic solid.
When using an AFM, care must be taken to ensure that these
deviations from linearity are not due to the common and
often ignored, nonlinear response of cantilever deflection
versus load for increasing loading conditions.80
With very small indentations, the working end of the
pyramid has been modeled as a sphere81especially when the
sharp tip has been made blunt.82–84More typically, pyra-
of cell indentation studies using AFMs.67,85We emphasize the
pyramidal indenter here, due to a citation error, which has
occurred when referencing the rigid cone solution. The solu-
tion of the pyramidal indenter as a rigid cone has been pre-
sented in numerous publications. Frequently, Sneddon’s 1965
publication78is cited for the solution of a rigid cone indenter.
However, the rigid cone solution published in Sneddon’s
1965 publication is not consistent with his previous publica-
tions75,77or the solution that preceded it.74Although this er-
ror has been noted before,86research articles continue to
improperly cite Sneddon’s 1965 article. This obviously be-
comes a problemwhen authors refer toSneddon’s 1965 article
without also including the equation they used. We therefore
recommend that Love74or Sneddon75be cited when refer-
encing the use of the rigid cone solution.
Methods that involve tensile stretching offer a more direct
and more economical approach for obtaining the material
polyacrylamide hydrogel with an overlay of the theoretic
force based on Equation IV in Table 1. The appropriate in-
dentation depth over which the fit was applied was
determined from a plot ofpF(1?v2)
2tan(a)d2¼E versus indentation
depth, (B), which demonstrates that the gel behaves as an
elastic solid up to *80nm of indentation.
(A) Indentation force versus indentation depth of a
Table 1. Review of Theoretical Models
ModelTheoretical F versus d
I Purely elastic sphere
II Rigid sphere indenter
III Rigid, flat-ended
IV Rigid cone indenter
V Rigid cone indenterF¼2
Note that the reduced modulus (E*) is used in Equation I and not
in II–V, where it is assumed that the indenter is infinitely rigid:
E, Young’s modulus; d, indentation depth; R, radius; n, Poisson’s
ratio; a, half angle opening.
YOUNG’S MODULUS OF SOFT TISSUES157
properties of a sample. Tests can be as simple as measuring
the change in length (strain) of a sample when a mass is
suspended from it (stress). Tensile measurements directly
quantify the strain that is induced by a given stress and under
the assumption of a linear elastic response, YM is determined
from the slope of the stress–strain curve (Equation 1).
Typical stress–strain relationships observed for tensile
measurements of soft biological tissues demonstrate that the
resistance to deformation of the tissue increases with in-
creasing stress. This nonlinear response means that the gra-
for E obtained by Equation 1 is always increasing. As men-
the initial linear response. Measurement of tensile stretch will
certainly lead to variation in reported values, as YM will be
dependent on the stress that is applied. More importantly
though, because the functional form of the stress–strain curve
is nonlinear and demonstrates increasing resistance to defor-
always increase with increasing range of fit. This leads one to
conclude that YM values measured by linear model fits using
tensile measurements for soft biological tissues arguably
represent an over estimate of the actual value. As noted by
Fung,51YM values, obtained by tensile measurements, must
be accompanied by a statement on the levels of stress and
strain applied to the tissue to be of any quantitative value.
Article inclusion criteria
The following are our inclusion criteria for values tabulated
in Tables 2 and 3: (1) Cited articles stated they measured the
elastic response of a tissue. (2) The cited articles used estab-
lished models for defining the elastic modulus, which is de-
pendent on tensile or indentation measurements. (3) To the
best of our ability, we confirmed that the reported values
corresponded with an elastic response. In the case of tensile
measurements, this was primarily determined by confirming
that the reported data displayed a linear stress–strain response
(although a number of articles also used nonlinear models). If
two elastic modulus values were reported, based on two
separate linear regimes, we used the smaller elastic modulus
value obtained at low strain. With the exception of ‘‘spinal
cord and gray matter,’’ if tensile articles presented both in-
stantaneous and relaxed elastic modulus values, we included
the lower, relaxed elastic modulus value. For ‘‘spinal cord and
gray matter,’’ all of the cited articles reported an instantaneous
modulus using a model solution for hyperelastic materials.87
For indentation measurements, if multiple YM values were
reported as a function of indentation depth, value inclusion
was limited to data reported from the initial response of the
sample. Review of indentation measurements has an addi-
tional complication in that the functional form of force versus
indentation curves for elastic materials is always nonlinear.
We therefore relied on representations of theoretical fits to the
raw data if it was presented. Not all articles included the raw
data that was used to define the reported value of YM, so we
also relied on the written wording of the article and criteria 1
and 2. (4) When possible, we ensured that for each group of
tissues, the cited articles were self-consistent in their mea-
surement. For example, in the methods sections of the tensile
reports on ‘‘tendon,’’ the authors described, in similar fashion,
that the gradient of stress–strain curves were measured di-
rectly after the ‘‘toe’’ region, in the ‘‘elastic phase’’ or ‘‘linear
region,’’ which they termed the elastic or YM of the sample.
This task was not always possible though, especially for in-
dentation measurements, as the model predictions are highly
specific to the indentation method used, in which case we
relied on criteria 1, 2, and 3. (5) We did not include YM values
from diseased tissues.
Comparison of Indentation and Tensile
Stretching Modulus Values
Tables 2 and 3 list YM values compiled from a number of
soft biological tissues measured by indentation and tensile
stretching, respectively. Tables 2 and 3 are arranged from
Table 2. Young’s Modulus of Soft Tissues,
Measured by Indentation
modulus (kPa) Reference
base & Descemet’s
& gray matter
L&K, liver & kidney; A&V, artery & vein.
Table 3. Young’s Modulus of Soft Tissues,
Measured by Tensile Stretching
Spinal cord &
158McKEE ET AL.
largest to smallest average modulus. YM values for inden-
tation have a range from *190kPa for the organs located in
the abdominal cavity, to around 3kPa for spinal cord and
gray matter. Tensile modulus values range from about
560MPa for tendons to *2.0MPa for spinal cord and gray
matter. Some reported modulus values appeared to be
clearly outside the median range. We suspect some of the
values are outliers and have tabulated averages for both
these outlier values included and excluded. For example, the
indentation of arteries and veins had reported YM values
that ranged from 6.5 to 21,000kPa, giving an average YM
around 3600kPa. However, the single reported value of
21,000kPa increases the average over 28-fold if included with
the other values, potentially complicating interpretation of
results. Table 4 compares the averaged result for the two
methods. Comparisons between indentation and tensile
measurements in Table 4 do no include the suspected outli-
ers noted in Tables 2 and 3.
To put these values in context, YM of a 25% aqueous so-
lution of gelatin is reported to be *30kPa88(sphere–sphere
compression), polydimethylsiloxane (PDMS) silicone rubber
*800kPa89(rheometry), and tissue culture polystyrene to be
*3 GPa90(indentation). In assembling this review, there
were some tissues that could not be directly compared due to
a lack of published YM values in either tensile or indentation
measurement. We have included these tissues to highlight
possible research areas of interest. Other materials of interest
to biological systems include polymeric hydrogels.91–95In
general, hydrogels can be formulated from a variety of
polymers with modulus values that span from very com-
pliant to extremely rigid. A particularly interesting biological
simulant, Matrigel?,31initially derived from a mouse tumor
cell line is used as a three dimensional platform for tissue
cultured cells and has a reported YM of around 1kPa.35
One can conclude from the information and data pre-
sented that the values of YM for soft biological tissues de-
pends on the method by which it is obtained. Additionally,
tensile measurements consistently result in larger YM values
compared with indentation measurements. The difference in
YM between these two methods is an experimental confir-
mation that soft biological tissues are not homogeneous over
all length scales. Indentation measurements are localized to
the region of indenter contact in the order of millimeters to
nanometers depending on probe size/geometry. Tensile
measurements induce macroscopic deformations that span
the bulk of a tissue with the entire specimen stretched. Un-
derstanding the differences is straightforward with muscles
or ligaments. These tissues are better suited to resist defor-
mation from a given tensile stress in the direction of fiber
orientation as compared to a localized indentation that might
be perpendicular to the fiber orientation. It is less clear,
however, why tensile measurements consistently result in
larger YM values for every soft tissue reviewed here. As
discussed earlier, the discrepancy may be due, in small part,
to the application of Equation 1 to tensile measurements of
tissues that display increasing resistance to deformation with
increasing stress, resulting in modulus values that are greater
than the actual YM. This may not account for the entire
difference, as tensile measurements of YM can be several or-
ders of magnitude greater than the indentation measurement.
One hypothesis that might contribute to the difference is due
to the combined response of extracellular matrices, individual
cells, longer-ranged protein polymers like collagen, actin and
elastin, and the effect of constrained water, which lead to an
increased resistance to deformation for macroscopic tensile
measurements. These effects would not be observed in in-
dentation measurements as the indenter induces sample de-
formation on the local environment only, both in terms of the
indenter geometry and the depth of penetration. This is par-
ticularly relevant in relation to the water constrained within
the interstitial spaces of the tissue. Indentations, especially in
the case of AFMs or nanoindenters, typically perturb the tis-
sue on the same scale as the constituents that make up the
tissue. The local volume of water around the indenter there-
fore contributes very little to the resistance to deformation, as
the tissue surrounding the indenter is under very little stress
and is therefore capable of accommodating these small fluc-
tuations in water content. Tensile measurements, on the other
hand, stress the bulk of all the constituents of the tissue.
Trapped water, which is incompressible, will significantly
increase the tissue’s resistance to deformation during an
applied tensile stress. The significant differences in reported
YM suggest that indentation and tensile measurements are
inherently measuring different properties of the same tissue
and that the scale of tissue perturbation is the dominant factor.
For example, a YM value that includes the effect of con-
strained water may best represent the behavior of a tissue,
in vivo, to a macroscopic load.
The strong dependence of YM on experimental method
can significantly affect our interpretation of tissue function,
disease status, or how cells respond to biophysical cues. A
clear distinction exists between macroscopic, microscopic,
and sub-microscopic responses to external stimuli. For ex-
ample, it has been shown that cellular behavior is influenced
by the compliance of the substrate on which the cell re-
sides.15,16,18,20,23,29,96–101To accurately relate these laboratory
observations back to in vivo cell function, compliance of the
tissue and the matrix with which a cell interacts must be
characterized. These values can be obtained with the AFM or
Table 4. Comparison of Indentation and Tensile
Measurements of Young’s Modulus
Indentation versus tensile
Tissue Indentation (kPa)Tensile (MPa)
Spinal cord & gray matter
The values are the averages without the suspected outliers. Of the
tabled citations on tissue mechanics, eight reports did not clearly
state the tissue hydration condition, and the rest were measured in
‘‘wet’’ or ‘‘hydrated’’ (such as skin) conditions. Of those eight, we
considered only one as an indentation outlier.109The results in this
table are therefore not changed by consideration of this variable.
YOUNG’S MODULUS OF SOFT TISSUES 159
possibly nanoindenters, but certainly not by macroscopic
tensile measurements. The design of larger implants such as
artificial joints or cartilage, however, may be better served by
macroscopic tensile measurements. The results of this review
clearly demonstrate that the rational design of engineered
tissues and biological simulants for use in the laboratory or
clinic must reflect the heterogeneous physical properties of a
soft tissue. Therefore, measurements of YM obtained by both
tensile and indentation methods are relevant in the design
and fabrication of such devices.
Of considerable interest is the modeling of tissue me-
chanics through mathematical models such as finite element
analysis, which is used to better understand how tissues may
respond to a given stress.102–108These models are complex
but can be understood generally by assuming that tissues
behave as perfectly elastic bodies and incorporate experi-
mentally obtained material properties, such as YM, to de-
scribe tissue function. As noted in the introduction, YM
values of soft tissues can be obtained only if one assumes
that the tissue behaves as an elastic body under small stress
conditions. However, the tabled data in this review dem-
onstrate that this assumption does not lead to a single YM
value independent of experimental method. At least two
values obtained from the different measurement techniques
exist for each soft tissue reviewed here (highlighting our
previous argument that the term YM must be very clearly
defined when describing the material property of a soft,
viscoelastic biological tissue). Therefore, research articles
using theoretical models must state how YM was deter-
mined, justify why a single value was used, or explicitly
account for the heterogeneous nature of soft tissues. For ex-
ample, it would not be appropriate to use tensile measures of
YM in mathematical models designed to understand tissue
mechanics at the micro- through nanoscopic scale.
The successful design of tissues engineered for improved
function or replacement of native tissues requires the com-
bined knowledge of biology, chemistry, and materials sci-
ence, as they must interface with complex biochemical and
biophysical environments in vivo. Increasingly, we under-
stand that the biophysical environment plays a crucial role in
the success of these engineered tissues. This review high-
lights and re-enforces an important attribute of soft tissues
that must be considered when trying to model or engineer
replacements; the response of soft tissues to an applied stress
should not be considered independent of experimental
method and that engineered tissues must reflect the hetero-
geneous material properties of native tissues.
In summary, this review has shown that soft biological
tissues do not have a single YM value independent of ex-
perimental method, and that modulus values for a single
tissue can span several orders of magnitude. If the method of
deformation is delineated then the data provided can be
better interpreted in context. On average, YM values of soft
tissues are consistently lower when obtained by local in-
dentation as compared to bulk tensile deformations. The
scientific objective of a given research proposal therefore
dictates which method is most appropriate.
The authors wish to thank the NIH for providing funding
for this review under the following grants: R01HL079012,
R01E4016134, R01EY19475, RC2AR058971, and P30EY12576.
Supported in part by an unrestricted grant from Research
to Prevent Blindness and a grant by National Glaucoma
Research, a program of the American Health Assistance
No competing financial interests exist.
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Address correspondence to:
Paul Russell, Ph.D.
Department of Surgical and Radiological Science
School of Veterinary Medicine
University of California Davis
1220 Tupper Hall
Davis, CA 95616
Christopher J. Murphy, D.V.M., Ph.D.
Department of Surgical and Radiological Science
School of Veterinary Medicine
University of California Davis
1220 Tupper Hall
Davis, CA 95616
Received: September 1, 2010
Accepted: February 7, 2011
Online Publication Date: March 10, 2011
164 McKEE ET AL.