The rigid skeletal system of vertebrates provides support and protection for soft tissues that reside within and on the skeletal frame. These soft tissues range from large organs to small connective tissues with cells and extracellular 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 macroscopic stiffness of tissues can indicate internal disease or injury. We now understand that changes in the bulk compliance of soft tissues can indicate the onset of conditions such as breast cancer,^{1–3} atherosclerosis,^{4–11} fibrosis,^{12} or glaucoma^{13} at a macroscopic level. At the micro- to nanoscopic level, bulk and local compliance influence a wide menu of fundamental cell behaviors, including cell morphology,^{14–17} proliferation,^{16,18–20} motility,^{15,21–25} differentiation,^{18,26–29} and response to therapeutic agents.^{30} It is therefore important to properly characterize the compliant biophysical state of tissues to understand how this biophysical 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 responses to biophysical stimuli, or the rational design and use of biological simulants.^{31–35} The 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 quantified compliant descriptor, Young's modulus (YM), by specifically 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 reported modulus values ranging from 2.9

kPa

^{36} to 19

MPa

^{37} when measured by atomic force microscopy (AFM)

^{38} tensile stretching,

^{39,40} tonometry,

^{41–43} or inflation/bulge testing,

^{36,37,44–47} This 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* measurements, tissue hydration state, time from death/tissue excision, temperature, storage medium, and the experimental method used. These experimental differences make direct comparison of results between studies difficult. Direct comparisons are also complicated by the various material properties 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.

^{48–50} Reliance 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 viscoelastic responses. YM is commonly used to try and quantify an intrinsic elastic property of soft, viscoelastic 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 complications have been discussed in detail^{51} and contribute to confusion and variability on reported mechanical properties of soft tissues. As noted by Fung,^{51} YM values obtained by tensile measurements must be accompanied by a statement 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 models^{39,51,52} that 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 measurement in N/m

^{2}.

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 (Δ

*L*/

*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 (nevertheless, nonlinear functions do not necessarily define a material as viscoelastic). For perfectly elastic materials a single YM value defines the response of material to deformation. For soft biological tissues, the resistance to deformation 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.

^{53} As 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 tissues 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 descriptions of tissue mechanics would make the stated purpose of this review impossible. However, the time dependent, viscous properties of biological tissues are important and we direct readers to the following reviews and articles, which discuss these properties for many of the tissues reviewed here.^{54–62} In 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 obtained 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).

Indentation

There are a variety of instruments used to indent samples,^{63–66} ranging from AFMs, capable of applying piconewton loads, to nanoindentors, which can resolve 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 requires 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.

^{67–70} YM values can also be obtained from the interpretation of unloading curves when the indenter is being withdrawn from the sample.

^{71} The 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 established.

^{72–78} lists the elastic model solutions for the common geometries used to determine YM. A full description of the elastic and viscous properties of a tissue would require additional measurement of its frequency-dependent response.

^{79} It is instructive to generalize the equations listed in to the following equation:

where

*F* is the force applied by the indenter,

*δ* is the indentation depth,

*α* 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 experimentally determined by controlling the indentation velocity and depth to produce loading and unloading curves that fall on one another. This result is not always possible as adhesions 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 predict a constant value of

*E* for any indentation 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, is a plot of indentation force versus depth on a polyacrylamide hydrogel prepared in our laboratory and indented using an AFM cantilever with an incorporated square pyramid tip. is a plot of

versus indentation depth, where

*F* is the force,

*ν* is Poisson's ratio,

*α* is the half angle opening of the AFM tip and

*δ* 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 ~80

nm of indentation. Beyond 80

nm,

*E* is no longer constant, indicating 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 sphere^{81} especially when the sharp tip has been made blunt.^{82–84} More typically, pyramidal indenters are assumed to be cones, especially in the case of cell indentation studies using AFMs.^{67,85} We emphasize the pyramidal indenter here, due to a citation error, which has occurred when referencing the rigid cone solution. The solution of the pyramidal indenter as a rigid cone has been presented in numerous publications. Frequently, Sneddon's 1965 publication^{78} is 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 publications^{75,77} or the solution that preceded it.^{74} Although this error has been noted before,^{86} research articles continue to improperly cite Sneddon's 1965 article. This obviously becomes a problem when authors refer to Sneddon's 1965 article without also including the equation they used. We therefore recommend that Love^{74} or Sneddon^{75} be cited when referencing the use of the rigid cone solution.

Tensile stretching

Methods that involve tensile stretching offer a more direct and more economical approach for obtaining the material 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 increasing stress. This nonlinear response means that the gradient of stress to strain is always increasing. Thus, the solution for

*E* obtained by

Equation 1 is always increasing. As mentioned earlier, YM values for soft tissues are typically based on 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 deformation, the gradient of linear fits (

*E*) to the initial response will 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,

^{51} YM 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 and : (1) Cited articles stated they measured the elastic response of a tissue. (2) The cited articles used established models for defining the elastic modulus, which is dependent 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 instantaneous 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 additional 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 measurement. 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 directly 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 indentation 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.

| **Table 2.**Young's Modulus of Soft Tissues, Measured by Indentation |

| **Table 3.**Young's Modulus of Soft Tissues, Measured by Tensile Stretching |