**Purpose.**:
To develop a mechanical model with which to investigate the relationship between the crimping morphology of collagen fibrils and the nonlinear mechanical behavior of the cornea.

**Methods.**:
Uniaxial tensile experiments were performed with corneal strips to test their mechanical behavior. A constitutive model was constructed based on the Gaussian-distributed morphology of crimped collagen fibrils. The parameters that represent the micro characteristics of collagen fibrils were determined by fitting the experimental data to the constitutive model. Transmission electron microscopy (TEM) was used to visualize the crimping morphology of collagen fibrils in the stroma. A quantitative analysis of fibril crimping degrees in the TEM images was conducted to test the parameters predicted by the constitutive model.

**Results.**:
The parameters were derived using a fitting method that included the expectation for the distribution of fibril crimping degrees, μ = 1.063; the standard deviation, *σ =* 0.0781; the elastic modulus of collagen fibrils, *E =* 52.74 MPa; and the fibril ultimate strain, *ε _{b} * = 0.1957. TEM images showed a variation of the fibril crimping morphology when the cornea was subjected to different tensile loads. A good agreement was found between the parameters derived by the constitutive model and the data quantified from the TEM images.

**Conclusions.**:
The nonlinear mechanical behavior of the cornea is closely correlated with the crimping morphology of collagen fibrils. The findings are expected to guide further research of corneal pathologies related to the abnormal microstructure of collagen fibrils.

^{ 1,2 }In the anterior stroma, collagen fibrils are strongly isotropic due to lamellar interweaving, whereas in the posterior stroma, most fibrils are arranged preferentially.

^{ 3–5 }X-ray scattering studies revealed that fibrils have a preferential orientation in the superior-inferior and nasal-temporal directions.

^{ 6–8 }Knowledge of the relationship between the organization of collagen fibrils and the mechanical properties of the cornea is important for understanding pathologies of corneal diseases related to abnormal microstructure such as keratoconus.

^{ 9–11 }

^{ 12 }Recently, based on the distribution and organization of collagen fibrils, several models have been introduced to describe corneal anisotropic behavior.

^{ 13–18 }Also, an inverse numerical analysis has been performed to determine mechanical parameters of the cornea by using an inflation test.

^{ 19 }A more recent study reported a constitutive model that, for the first time, accounts for three-dimensional, inclined lamellae and their depth-dependent distribution.

^{ 20 }

^{ 21,22 }proposed a constitutive model to investigate the interaction between the fibril morphology in the stroma and the mechanical conditions of the cornea. This model was based on the hypothesis that collagen fibrils in a load-free state are crimped like a helical spring.

^{ 23,24 }The spring, under a tensile force, was used to represent mechanical behavior of the fibrils. Studies related to fibril crimping morphology were also conducted by Lanir,

^{ 25,26 }Zulliger et al.,

^{ 27 }Hurschler et al.,

^{ 28 }and Cacho et al.

^{ 29 }

^{ 30 }It can diminish the mechanical interference caused by the fibrils in nonpreferential directions. Twenty-four enucleated bovine eyes were obtained from a local abattoir within 4 hours postmortem. The entire cornea with adjacent sclera (∼2–3 mm width) was then extracted from each eye (Fig. 1). To maintain corneal hydration without swelling, the specimens were preserved in Eusol C (Alchimia, Padova, Italy) within 4 hours before the mechanical tests. An experimental test was conducted to analyze corneal swelling in different solutions (see Figure S1 in Supplement 1). The result indicated that the corneas showed no swelling when they were kept in Eusol C during that time, which is also supported by previous studies.

^{ 37,38 }Using a self-designed double-bladed scalpel, we obtained a 2-mm-wide limbus-to-limbus strip centered on the cornea, along the inferior-superior direction, from each specimen. Mechanical tests were conducted using a material testing machine equipped with a load cell capable of 50 newtons (N; 800LE mechanical test system; TestResources, Inc., Shakopee, MN, USA). The strip was connected to a pair of clamps with a length of approximately 3 to 4 mm on both sides, so that only the central region of the strip was tested. To prevent dehydration, corneal strips were kept in a water bath system (BioBath environmental chamber; TestResources) filled with Eusol C solution. Uniaxial tensile tests were performed with the corneal strips with a speed of 1 mm/min until the breaking point was reached. A schematic representation of the uniaxial tensile test is shown in Figure 1. As the tests were conducted in solution, it is important to note the buoyancy effect, which would influence the measurements. To obtain more accurate data, a nonload test in solution was conducted before each uniaxial test. The final results eliminated the buoyancy effect by subtracting the nonload measurements from the load elongation measurements.

**Figure 1**

**Figure 1**

^{ 21,22 }Accordingly, a constitutive model was introduced to describe the fibril morphological change under different load conditions. The modeling was based on an important assumption: the cornea is composed of collagen fibrils that lie only in the preferential (tensile) direction. This assumption is supported by studies by Meek et al.

^{ 3,6 }that show collagen fibrils in the posterior stroma are arranged preferentially. Contributions by other components and the interaction between the neighboring fibrils are not considered in the model.

*d*, which is defined as the ratio between the straightened length

_{c}*l*and the crimped length

_{s}*l*of a fibril:

_{c}*d*=

_{c}*l*/

_{s}*l*. When the corneal strip is stretched, a single fibril experiences 3 phases: a crimped, straightened, and broken phase (Fig. 2). As the fibril is crimped, it is unable to resist the tension due to its slack shape. With elongation of the corneal strip, the fibril crimping degree decreases gradually. When the fibril is straightened, it has the capability of tension resistance and this contributes to the mechanical reinforcement of the cornea. The straightened fibril follows Hooke's law. Once the load on the fibril exceeds the tensile strength,

_{c}*ε*(maximum strain), the fibril will break immediately and never resist any tension (Fig. 2). The relationship between the stretch of a corneal strip, Λ, and the tensile stress of a single fibril,

_{b}*t*, can be written as: where the stretch parameter Λ is defined as the ratio between the elongated length,

*L*, and the initial length,

*L*, of the corneal strip. Variable

_{0}*e*is the elastic modulus of the fibril. More details of the mathematical development of Equation 1 can be found in Appendix A.

**Figure 2**

**Figure 2**

*σ*are called expectation and standard deviation, respectively, which together represent the characteristics of a Gaussian distribution.

**Figure 3**

**Figure 3**

*T*, can be given as an integral of the fibril crimping degrees: where

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

^{ 31 }: the O-A region (“toe” region) is characterized by stress increasing nonlinearly with stretch, where

*σ*

_{A}= 1.492 ± 0.431 MPa, and

*λ*

_{A}= 1.096 ± 0.017. The A-B region shows a strong linearity of the stress-stretch curve, whose slope is the elastic modulus of the cornea, where

*E*= 34.113 ± 6.14 MPa. Point B is the endpoint of the linear region, where

*σ*

_{B}= 4.895 ± 0.860 MPa, and

*λ*

_{B}= 1.226 ± 0.028. The B-C region, named the “heel region,” presents a nonlinearly shaped area. At point C, the corneal strip ruptured, where

*σ*

_{C}= 5.513 ± 0.907 MPa, and

*λ*

_{C}= 1.224 ± 0.033, which is the corneal tensile strength.

**Figure 6**

**Figure 6**

*σ*, for the distribution of fibril crimping degrees, the elastic modulus

*E*, and the ultimate strain,

*ε*, of collagen fibrils. The expectation value, μ = 1.063, indicates that most collagen fibrils appear to have a crimping degree of 1.063 under a load-free condition. The standard deviation,

_{b}*σ*, represents the amount of dispersion away from the expectation. The parameters μ and

*σ*together determine the shape of the probability density curve of the fibril crimping degrees (see Fig. 9). Note that the density curve is asymmetrical as it is redundant when

*d*is ≤1. Thus, the value of the standard deviation,

_{c}*σ*, provides an approximation of the data dispersion.

*E*and

*ε*represent the mechanical properties of collagen fibrils, which were determined by the characteristics of the corneal stretch-stress curve. The elastic modulus of the collagen fibrils was predicted to be 52.74 MPa, which is less than the order of magnitude (1.0 GPa) reported in previous studies.

_{b}^{ 32 }In the constitutive model, the cornea strip is assumed to be fully composed of collagen fibrils. Therefore, the elastic moduli of the cornea and collagen fibrils are of the same order. The macro mechanical properties of the cornea have a strong influence on the parameters

*E*and

*ε*.

_{b}**Figure 7**

**Figure 7**

**Table**

**Figure 8**

**Figure 8**

**Figure 9**

**Figure 9**

**Figure 10**

**Figure 10**

^{ 33 }In the stroma, collagen fibrils are surrounded by a matrix of water and proteoglycan, which contributes little to the loading resistance in the strip test.

^{ 39 }Effects of the ground matrix and its interaction with the collagen fibrils were not considered in the current study. Therefore, the preferential collagen fibrils in the stroma were primarily responsible for the mechanical behavior of the corneal strip. However, it should be noted that the hydrated matrix mainly determines the viscous (time-dependent) behavior of the cornea. Hydration and proteoglycan distribution should be accounted for when modeling corneal viscoelastic behavior.

^{ 31 }: the “toe” region, the linear region, and the “heel” region. In the “toe” region, the corneal stiffness (elastic modulus) grows quasi-exponentially with elongation. The TEM image shows that collagen fibrils exhibit a wavy appearance with high crimping degrees in the load-free strip (Fig. 8A). With the corneal strip elongated, the fibril crimping degrees decreased until the fibrils are straightened. When the fibril crimping degrees were presumed to follow a Gaussian distribution, the constitutive model could well reproduce the corneal stress-stretch curve in the tensile test (Fig. 7). This confirmed that the distributed crimped morphology of the collagen fibrils is responsible for the nonlinear behavior of the cornea, despite the fact that individual fibrils are linearly elastic.

^{ 34–36 }Previous studies suggested that the corneal elastic modulus obtained by strip extensometry test is significantly larger than that obtained by inflation test.

^{ 37,40 }Also, Elsheikh and Anderson

^{ 41 }introduced a model to match the parameters obtained from both methods. However, even though the corneal elastic moduli were derived from strip extensometry tests, there are significant differences among the measurements. Many factors can be responsible for the large dispersion of the values, including the specimen origins, the level of strain, the testing protocol, and the calculation method. Hoeltzel et al.

^{ 42 }and Zeng et al.

^{ 34,43 }concluded that there is little difference in the mechanical properties of different origins. However, corneal elastic modulus under a large strain (∼10–60 MPa) is much greater than that under a small strain (∼0.1–3 MPa).

^{ 37,44 }The experimental protocols such as the preservation solution,

^{ 45,46 }the cycle number of the preloading,

^{ 37,41 }and the testing environment (e.g., use of water bath or not)

^{ 43,47 }make a great difference in the results. Some corneal elastic moduli in the literature are derived using a secant modulus, which is less than a tangent modulus in the linear region of the strain-stress curve.

^{ 37,48 }Also, some results in the literature were obtained from a modified formula to overcome the effects of nonphysiological loading.

^{ 44,49 }

**X. Liu**, None;

**L. Wang**, None;

**J. Ji**, None;

**W. Yao**, None;

**W. Wei**, None;

**J. Fan**, None;

**S. Joshi**, None;

**D. Li**, None;

**Y. Fan**, None

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*l*and

_{c}*l*are the initial crimped length and straightened lengths, respectively, of the fibril. The crimping degree is the ratio of

_{s}*l*to

_{c}*d*, where

_{c}*l*/

_{s}*l*is the length when the fibril is stretched just to the breaking point.

_{c·}l_{b}*ε*is the maximum tensile train that a single fibril can bear, which can be written as:

_{b}*ε*. Accordingly, when a corneal strip is stretched to

_{b}*L*, the relationship between the elongation of the strip

*L*and the tensile stress on a single fibril,

*t*, can be written as: where

*e*is the elastic modulus of a fibril. Lowercase letters are used for parameters related to a single fibril at the micro level, whereas capital letters are used for parameters of a corneal strip at the macro level.

*L*and the initial length of a corneal strip: Λ =

*L*/

*L*

_{0}. It should be noted that

*L*

_{0}is equal to the initial crimped length of a single fibril:

*L*

_{0}=

*l*. Then, the Equation A2 can be rewritten as:

_{c}*f*(

*d*,

_{c}*σ*, μ) (Equation 2). The graph of Gaussian probability represents a typical bell-shaped curve, which is characterized by 2 parameters: the mean

*μ*and the standard deviation

*σ*(Fig. 1A). When a corneal strip is stretched to Λ, the collagen fibrils of

*d*= Λ are just straightened. Without considering fibril break, the tensile stress produced in the cornea,

_{crimp}*T*, is carried by all the straightened fibrils, (

*d*≤ λ). Then, the integral ranges from 1 to Λ, which can be expressed as:

_{crimp}*d*< Λ/(1 +

_{c}*ε*) have broken and lost their load resistance. Accordingly, the lower limit is mortified as Λ/(1 +

_{b}*ε*). It should be noted that when Λ < 1 +

_{b}*ε*, the lower limit returns to 1 because the Λ is never less than 1.

_{b}