**Purpose.**:
The authors applied a novel microindentation technique to characterize biomechanical properties of small ocular and orbital tissue specimens using the Hertzian viscoelastic formulation, which defines material viscoelasticity in terms of the contact pressure required to maintain deformation by a harder body.

**Methods.**:
They used a hard spherical indenter having 100 nm displacement and 100 μg force precision to impose small deformations on fresh bovine sclera, iris, crystalline lens, kidney fat, orbital pulley tissue, and orbital fatty tissue; normal human orbital fat, eyelid fat, and dermal fat; and orbital fat associated with thyroid eye disease. For each tissue, stress relaxation testing was performed using a range of ramp displacements. Results for single displacements were used to build quantitative Hertzian models that were, in turn, compared with behavior for other displacements. Findings in orbital tissues were correlated with quantitative histology.

**Results.**:
Viscoelastic properties of small specimens of orbital and ocular tissues were reliably characterized over a wide range of rates and displacements by microindentation using the Hertzian formulation. Bovine and human orbital fatty tissues exhibited highly similar elastic and viscous behaviors, but all other orbital tissues exhibited a wide range of biomechanical properties. Stiffness of fatty tissues tissue depended strongly on the connective tissue content.

**Conclusions.**:
Relaxation testing by microindentation is a powerful method for characterization of time-dependent behaviors of a wide range of ocular and orbital tissues using small specimens, and provides data suitable to define finite element models of a wide range of tissue interactions.

^{ 1 –3 }crystalline lens,

^{ 4 –6 }sclera,

^{ 7 –10 }iris,

^{ 11,12 }orbital fat,

^{ 13,14 }orbital connective tissue,

^{ 14,15 }and extraocular muscle (EOM).

^{ 16 –18 }However, idiosyncratic techniques used in some tissues often lead to results that cannot be generally compared among tissues.

^{ 19 –22 }a general mathematical formulation that describes material stiffness in terms of the contact pressure to initiate deformation by a harder spherical body.

*P-t*) data of two trials were averaged for each displacement distance. At least four displacement distances were used to compute a viscoelastic model for each tissue type.

*R*, to a depth,

*h*, by a ramp trajectory within the framework of Hertzian contact mechanics

^{ 21,23 }that is derived fully in the Appendix. Briefly, a load-relaxation function consisting of three exponential terms was first fit to the experimental data using the Levenberg-Marquardt nonlinear least squares method.

^{ 21 }Material parameters,

*C*, were determined from force-relaxation parameters,

_{i}*B*. The model incorporates a ramp correction factor (RCF) that accounted for finite, rather than ideally instantaneous, ramp loading time. Short-term (

_{k}*E*

_{0}) and long-term (

*E*

_{∞}) elastic stiffnesses are generalized biomechanical descriptors of the tissues and can be determined from the fitted viscoelastic model with material parameters. These elastic stiffnesses constitute the primary outcome measures of the study.

^{ 24,25 }whereas other authors have reported a single stiffness value.

^{ 26,27 }Because of the size of the spherical indenter and the curved shape of the lens, only the center of the lens was tested. Examinations after indentations were performed to verify that the lens capsules remained intact.

^{ 14 }Ten pulley tissue specimens extracted from the previously determined pulley and 10 fatty tissue specimens taken from the orbit between the retractor bulbi and rectus EOMs

^{ 14 }were prepared from four orbits. Specimens were cut to a 5-mm-diameter at 3-mm thickness.

^{ 14 }Samples were dehydrated in alcohol solutions with different concentrations from 50% to 100%. After complete dehydration, the fatty portion of the tissue was dissolved in xylene. Tissue samples were then serially sectioned at a 5-μm thickness after they were embedded in paraffin and stained with hematoxylin. Photographs in 24-bit color were made using a microscope (E800N; Nikon, Tokyo, Japan) fitted with a digital camera (D1X; Nikon). Four randomly selected micrographs were thresholded into binary images; the number of white pixels representing eluted fat was compared with the total number of tissue pixels to determine the percentage of fat in each specimen.

*t*, of 13.7 seconds. Viscoelastic model parameters were determine by fitting to the lower strain rate data and were used to predict the response for the higher ramp displacement. There was excellent agreement between the model and the data because the value for maximum error between the model prediction and the data were 4.9% at the higher ramp loading rate. These findings indicate general validity of the Hertzian approach with data from the custom load cell.

_{r}*R*

^{2}) values for all five ramp displacements exceeded 0.98 for both inner and outer scleral surfaces. Short- and long-term stiffnesses computed from the viscoelastic models were 119 ± 2.6 (± SEM), and 27.5 ± 2.5 KPa, respectively, for the inner scleral surface and 31.9 ± 1.6 and 17.0 ± 0.3 KPa, respectively, for the outer scleral surface (Table 1). Although larger ramp displacements were used for the outer than for the inner scleral surface, resultant peak forces were lower for the outer sclera, indicating that the inner scleral surface is less elastic than the outer scleral surface. In addition, both average stiffnesses for the outer scleral surface (

*n*= 5) were significantly smaller than the stiffnesses for the inner scleral surface (

*t*-test;

*P*≤ 10

^{−8}for short-term and

*P*≤ 10

^{−5}for long-term stiffness).

Bovine Outer Sclera | Bovine Inner Sclera | Bovine Lens | Bovine Iris | Bovine Orbital Connective Tissue | Bovine Orbital Fat | Human Orbital Fat (thyroid disease) | Human Eyelid Fat | Human Orbital Fat (normal) | Bovine Kidney Fat | Human Dermal Fat | |
---|---|---|---|---|---|---|---|---|---|---|---|

B _{0}, g/mm | 0.28 | 0.35 | 1.29 | 0.03 | 0.13 | 0.27 | 0.19 | 0.15 | 0.47 | 0.02 | 0.06 |

B _{1}, g/mm | 0.13 | 0.58 | 36.1 | 0.22 | 0.11 | 0.12 | 0.24 | 0.30 | 4.15 | 0.48 | 0.02 |

τ_{1}, s | 10.2 | 7.79 | 1.51 | 3.01 | 22.5 | 34.4 | 12.91 | 13.8 | 2.04 | 2.41 | 31.7 |

B _{2}, g/mm | 5.06 | 1.45 | 0.24 | 0.16 | 0.20 | 0.17 | 0.14 | 9.99 | 18.6 | 0.05 | 0.02 |

τ_{2}, s | 1.01 | 1.56 | 1.76 | 23.9 | 3.29 | 23.5 | 111 | 1.52 | 2.86 | 9.66 | 31.7 |

B _{3}, g/mm | 0.09 | 0.58 | 0.77 | 0.53 | 0.56 | 6.72 | 1.85 | 0.13 | 0.62 | 0.05 | 0.37 |

τ_{3}, s | 19.40 | 37.5 | 23.4 | 3.01 | 0.65 | 3.33 | 1.41 | 105 | 48.5 | 84.6 | 2.37 |

RCF _{1} | 1.36 | 1.40 | 6.81 | 1.65 | 1.03 | 1.09 | 1.10 | 1.22 | 10.7 | 6.20 | 1.11 |

RCF _{2} | 65.5 | 7.38 | 4.97 | 1.06 | 1.21 | 1.14 | 1.01 | 9.53 | 4.92 | 1.48 | 1.11 |

RCF _{3} | 1.17 | 1.07 | 1.11 | 1.65 | 2.88 | 2.87 | 2.56 | 1.03 | 1.08 | 1.04 | 5.11 |

C _{0}, g/mm | 1.16 | 1.87 | 0.23 | 0.02 | 0.22 | 0.03 | 0.08 | 0.03 | 0.04 | 0.05 | 0.27 |

C _{1}, g/mm | 0.39 | 2.23 | 1.26 | 0.07 | 0.16 | 0.02 | 0.09 | 0.04 | 0.04 | 0.08 | 0.08 |

τ_{1}, s | 10.2 | 7.79 | 1.51 | 3.01 | 22.5 | 34.4 | 12.91 | 13.8 | 2.04 | 2.41 | 31.7 |

C _{2}, g/mm | 0.32 | 1.05 | 0.01 | 0.08 | 0.22 | 0.02 | 0.08 | 0.18 | 0.26 | 0.03 | 0.08 |

τ_{2}, s | 1.01 | 1.56 | 1.76 | 23.9 | 3.29 | 23.5 | 111 | 1.52 | 2.86 | 9.66 | 31.7 |

C _{3}, g/mm | 0.30 | 2.93 | 0.16 | 0.17 | 0.41 | 0.29 | 0.38 | 0.02 | 0.04 | 0.05 | 0.30 |

τ_{3}, s | 19.4 | 37.54 | 23.4 | 3.01 | 0.65 | 3.33 | 1.41 | 105 | 48.5 | 84.6 | 2.37 |

E _{0}, KPa | 31.9 | 119 | 25.6 | 4.86 | 18.5 | 7.91 | 11.2 | 5.90 | 7.86 | 2.68 | 10.6 |

E _{∞}, KPa | 17.0 | 27.5 | 4.50 | 0.24 | 3.99 | 0.74 | 1.71 | 0.48 | 0.71 | 0.34 | 3.92 |

*R*

^{2}values consistently exceeding 0.97.

^{2}, whereas the minimum

*R*

^{2}between model prediction and data for all lens specimens was 0.97, showing excellent agreement. Behavior predicted from the viscoelastic mechanical model based on data from one displacement accurately described data for all other displacements tested in each specimen. Observed short-term and long-term stiffnesses in the bovine lens were 25.6 ± 3.5 (± SEM) and 4.50 ± 1.60 KPa, respectively.

^{2}, respectively, whereas

*R*

^{2}values ranged from 0.97 to 0.99. The short-term and long-term elastic moduli for bovine iris specimens were 4.86 ± 0.38 (± SEM) and 0.24 ± 0.16 KPa, respectively.

*R*

^{2}values exceeded 0.95. Computed short-term and long-term stiffnesses from the model were 18.5 ± 1.8 (± SEM) and 3.99 ± 1.84 KPa, respectively, for bovine orbital connective tissue and 7.91 ± 0.41 and 0.74 ± 0.08 KPa, respectively, for bovine orbital fatty tissue. Orbital connective tissue was, therefore, stiffer than orbital fatty tissue by at least fivefold (

*P*≤ 10

^{−4}for short-term stiffness and

*P*≤ 10

^{−7}for long-term stiffness).

*P*< 10

^{−3}for short-term stiffness and

*P*≤ 10

^{-−4}for long-term stiffness).

*P*> 0.3 for short-term stiffness and

*P*> 0.1 for long-term stiffness). Bovine kidney fat was only approximately one-third as stiff as orbital fat, with short- and long-term stiffnesses of 2.68 ± 0.61 and 0.34 ±0.24 KPa, respectively. Both stiffnesses of bovine kidney fat differed significantly from those of human orbital fat (

*P*≤ 10

^{−5}for short-term stiffness and

*P*≤ 0.002 for long-term stiffness). On the other hand, short-term stiffnesses of human dermal fat were twice those of lid fat at 10.6 ± 1.0 KPa (

*P*≤ 10

^{−5}), and the long-term stiffness of human dermal fat was nearly 10-fold higher than that of orbital fat at 3.92 ± 0.14 KPa (

*P*≤ 10

^{−7}).

^{ 23,28 }Hertzian models based on parameters fit to a subset of microindentation data accurately predicted behavior over a wide range of indentation amplitudes because the coefficients of determination of model predictions and experimental results for all 11 tissues averaged 0.97. This verifies that the Hertzian model is an excellent and general description of viscoelastic behavior of ocular tissues.

^{ 29 }estimated the elastic modulus, which is equivalent to long-term stiffness in the present study, for intraconal and extraconal retrobulbar fat to be 0.3 and 1.0 KPa, respectively. Although “muscle cone” as a demarcating structure in the deep orbit is an anatomic fiction,

^{ 30 }the bovine orbital fatty tissue in current investigation may correspond to what Schutte et al.

^{ 29 }termed intraconal retrobulbar fat. The present study determined the long-term stiffness of human orbital fat to be 0.71 KPa, which is of similar magnitude to that reported by Schutte et al.

^{ 29 }A more important observation of the current investigation, however, was similarity in the stiffness of normal human and bovine orbital fatty tissue. As can be seen in Table 1, the long-term stiffness of human orbital fatty tissue and bovine orbital fatty tissue are similar at 0.74 and 0.71 KPa, respectively. Although there is <5% difference in long-term stiffness between human and bovine orbital fatty tissues, dense bovine connective tissue from around the globe equator has a much higher long-term stiffness of 3.99 KPa, which is within same order of magnitude of stiffness for extraconal retrobulbar fat reported by Shutte et al.

^{ 29 }The long-term stiffness of dense orbital connective tissue, also known as pulley tissue, exceeds that of orbital fatty tissue close to threefold, which agrees with the assessment of Yoo et al.

^{ 14 }Highly stiff connective tissue, of which pulleys are composed, must be therefore be distinguished in biomechanical models from the much less stiff orbital fat. Pulley tissue is highly collagenous (Fig. 6), making its mechanical properties markedly different from those of orbital fatty tissue.

^{ 31,32 }there have been no previous in vitro measurements of elastic stiffness of orbital fat in this disorder. As seen in Figure 5, orbital fat in thyroid eye disease has significantly higher short- and long-term stiffness values at 11.2 and 1.7 KPa than those of normal human and bovine orbital fatty tissues. However, both stiffnesses are approximately half that of bovine orbital connective tissue, clearly suggesting that connective tissue content makes a more significant contribution to the structural stiffness of soft tissue than does fat, which is infiltrated by fibrosis in thyroid eye disease.

^{ 7 }determined the static compressive modulus of cattle sclera range from 12 to 19 KPa, which is comparable to our long-term elastic modulus for inner sclera of 27.5 KPa. Using uniaxial tensile testing, Downs et al.

^{ 9 }found long-term elastic moduli of 7.46 ± 1.58 MPa for monkey eyes with early glaucoma and 4.94 ± 1.22 MPa for healthy monkey eyes. It thus appears that that scleral tissue may be 50-fold stiffer under tensile than under compressive loading.

^{ 11 }previously reported the radial stiffness of the iris sphincter and dilator to be 340 KPa and 890 KPa, respectively. This should not be expected to be comparable to the compressive loading data reported here. Elastic moduli for bovine iris tissue resulting from the viscoelastic model in current investigation are 4.86 and 0.24 KPa, indicating that stiffness in the radial direction is different from transverse stiffness. For bovine lens, short-term and long-term elastic moduli were found here to be 25.6 and 4.50 KPa, respectively. Fisher

^{ 4 }reported that the stiffness of the human lens is much lower, ranging from 0.75 to 3.0 KPa. Although there might be discrepancy between the mechanical properties of human and bovine lens, the long-term stiffness of bovine lens found in the present study certainly falls within the same order of magnitude.

*R*, is pressed a distance,

*h*, into an incompressible material with elastic modulus,

*E*, the sphere exerts force,

*P*, that is a function of Poisson ratio ν, the ratio of the transverse contracting strain to the elongation strain (equation 1a). If the material is incompressible, as seems reasonable for biological materials, the Poisson ratio is, by definition, ν = 0.5, and the shear modulus

*G*=

*E*/3. shows that the indenting force,

*P*, is a nonlinear function of indenter displacement and a linear function of shear modulus.

^{ 33 }Instead of constants, a viscoelastic operator is substituted for the elastic modulus,

*E*, in equation 1a, or shear modulus,

*G*, in equation 1b.

^{ 21,22,28,33 }The elastic modulus and relaxation functions (represented as a Prony series, which is a sum of exponential functions) then can be fit empirically.

^{ 21,23,28 }

^{21,22,28}where

*G*(

*t*) is the time-dependent shear relaxation modulus. Given that ideal instantaneous step loading is not physically attainable, actual rise time (

*t*) for ramp loading should be considered in the derivation.

_{R}^{21,23}Equation 3 is a viscoelastic integral operator for relaxation where

*u*is a strain function of time dummy variable τ. As suggested by Mattice et al.,

^{21}a Boltzmann integral method

^{23}is used here. When the time-dependent relaxation modulus in equation 2 is combined with equation 3, the resultant Boltzmann integral equation is shown in equation 4. For ramp-loading rate

*k*, displacement for ramp-hold relaxation can be written as The solution for displacement-controlled relaxation is expressed as the step-loading relaxation solution adjusted by an RCF to take into account the difference in relaxation caused by noninstantaneous ramp loading.

^{23,28}Since load

*P*(

*t*), is exponentially decaying during relaxation, it is expressed as whereas the material relaxation function has the form, where τ represents each time constant for each exponential form,

*B*represents a fitting constant, and

_{n}*C*represents relaxation coefficients. It is parsimonious and, therefore, desirable to minimize the number of exponential terms for curve fitting. We settled on three terms that captured relaxation behavior with <6.5% error. Once all the fitting parameters (

_{n}*B*) have been determined, they can be converted to material parameters (

_{n}*C*) using equations 9 and 10. The equation for RCF, which compensates for actual ramp versus ideal step loading, is shown in equation 11. Instantaneous

_{n}*G*(0) and long-time

*G*(∞) stiffnesses can be computed from the fitted relaxation coefficients (

*C*), as shown in equations 12 and 13.

_{i}