An Indentation Technique to Characterize the Mechanical and Viscoelastic Properties of Human and Porcine Corneas

Abstract

Cornea is a load-bearing tissue whose mechanical and viscoelastic characteristics are not well understood, due to the challenge associated with most of the measurements. A novel indentation technique has been developed for mechanical characterization of human and porcine corneal tissue, using a tailored depth-sensing microindentation instrument. During indentation, the corneas were suspended by clamping the edges of the cornea, thus allowing depth-sensing measurement free from the complication of the backing substrate. The deformation displacement and the amount of force applied by the indenter were used to obtain hysteresis and stress relaxation data for both human and porcine corneas. Optical coherence tomography was used to measure the thickness of the cornea. Simple theoretical analyses have been undertaken to explain the loading–unloading and the stress relaxation data. The effect of swelling on the mechanical properties of the cornea was also examined. Porcine corneas appeared to be less stiff and to demonstrate more linear response than human corneas under loading. More importantly, it is shown that swelling reduced the strength of the corneas. Our results demonstrate that this new indentation system can be used to characterize the mechanical and viscoelastic properties of corneas.

Appendix

A diagram depicting the theoretical model of an indented cornea is shown in Fig. A1. In order to determine the relationship between the applied force and the strain, the cornea is assumed to be isotropic, parabolic in shape, and have a uniform thickness. The profile of the cornea can described as a line by the equation,

$$ y=ax^{2}+bx+c , $$

(A1)

where a, b, and c represent constants. The slope at any point along the line can be found from the equation,

$$ \frac{dy}{dx}=2ax+b . $$

(A2)

By substituting the points (r,0), and (0,w) into Eq. (A1) and (0,0) into Eq. (A2), the values for a, b, and c can be written as

$$ a=\frac{-w}{r^{2}}, $$

(A3)

$$ b=0, $$

(A4)

$$ c=w, $$

(A5)

where w is the vertical height of the clamped cornea and r is the radius at the clamped portion of the cornea. Equation (A1) can then be rewritten as follows,

$$ y=w\left( 1-\frac{x^{2}}{r^{2}} \right). $$

(A6)

The distance L from the clamped edge of the cornea to the center of the cornea before indentation, can be found from the equation

$$ L=\int\limits_0^r (1+4a^{2}x^{2})^\frac{1}{2}dx. $$

(A7)

At an indentation depth of δ, the strain, ɛ, can be assumed to be

$$ \varepsilon =\frac{L^{\prime}-L}{L}, $$

(A8)

where L′ is the distance from the clamped edge of the cornea to the center of the cornea during indentation and can easily be determined by replacing w with w + δ in Eq. (A3). The force applied to the cornea by the indenter can be considered to be equivalent to a point load as the indenter radius is significantly smaller than the radius of the clamped portion. The force, F, applied to the cornea can be described as

$$ F=2\pi hr \sigma {\sin}\theta , $$

(A9)

where σ is the resulting stress around the clamped portion of the cornea, θ is the angle between the cornea edge and the horizontal, and h is the cornea thickness. θ can be determined from the equation

$$ \theta =\hbox{tan}^{-1}(2ar) . $$

(A10)

It is generally accepted that most biological materials including cornea exhibit a non-linear viscoelastic relationship between stress and strain and therefore a single value for the elastic modulus cannot be accurately obtained.27 Hoeltzel et al.11 used the equation

$$ \sigma =\alpha (\varepsilon -\varepsilon_{\rm s})^{\beta} $$

(A11)

to define the relationship between tensile stress and strain where α represents a scaling factor, β represents the non-linear component between stress and strain and ɛ_s is the slack strain (the lowest strain to result in an increase in stress). By combining Eqs. (A9) and (A11), the relationship between force and strain can be described by the equation

$$ F=2\pi hr\alpha (\varepsilon - \varepsilon_{\rm s})^{\beta}{\sin}\theta $$

(A12)

which is equivalent to Eq. (1) in the text. This model has a number of limitations. One limitation of this model is the assumption that the cornea is isotropic rather than anisotropic. We would expect an anisotropic material to deform differently to an isotropic material and therefore assuming isotropy will lead to an error in our results. This error is dependent on the structure and deformation behavior of the cornea. Another limitation may be the assumption that the cornea undergoes stretching rather than bending. The relationship between bending and stretching for flat membranes during indentation has been previously examined.4,29 The relationship for a flat circular membrane can be described using the equation4

$$ \lambda =\left[ 12\left( 1-\nu^{2} \right)\right]^{\frac{3}{2}}\left( \frac{Fr^{2}}{Eh^{4}} \right), $$

(A13)

where λ is a dimensionless parameter to determine whether bending can be ignored, ν is the Poisson’s ratio, and E is the elastic modulus. The values obtained for λ using this equation ranged from 200 to 2000, which fall within the bending–stretching transition region for a flat circular membrane4 (85 < λ < 30,000). This suggests that both bending and stretching influence the deformation behavior of the cornea. However, since this equation has been derived for a flat material it does not accurately represent the deformation of the cornea, which should be less influenced from bending due to its already curved structure. Despite these limitations the present model provides a simple first order analysis of the material properties and serves the purpose of mechanical characterization. Further work to improve the model and predict the relationship between stretching and bending during indentation would be required to obtain more accurate results.

Figure A1

Schematic of theoretical model for portraying cornea before and during indentation