Dear Imran,

theoretical formulation for cornea deflection is probably not a good idea, because you will be forced to made a lot of simplifications.

There are two ways:

1. measurement, and simple curve fit approach to resemble a experimental data

2. precise Finite Element model tuned by experimental data

First way is simple. Second one is quite difficult because it needs material properties data, which are itself nonlinear and anisotropic. But if successful done, will let you to understand what is going on during cornea deflection.

Surely this object in nonlinear, even for small deflection, for a few reasons:

1. material is nonlinear

2. shape of the cornea cause very significant nonlinear response because of so called geometrical nonlinearity - and this is a main problem. In short, the small change in such geometry cause the big change in stiffness, even for linear materials.

3. the external support is on corneal limbus is nonlinear.

I hope it helps a little.

Dear Artur,

Thank you so much for your reply, I greatly appreciated. You are right, however the curve fit itself is also quite tricky if I try to find the elastic modulus of the cornea precisely. I am now trying to fit my experimental data to a third order polynomial function (y=Ax+Bx^3, x is deflection, y is load) which is originally derived for large deflections of thin plates. I thought this is a good assumption since the first term corresponds bending, and the second term corresponds to stretching of the shell. The fit looks reasonable, however the elastic modulus that I obtain using B term does not make sense (too big). May be the formulation of the B term (how it is related to the elastic modulus) that I am using is not true, that's why I wanted to derive a more accurate formula for the cornea or at least have a reference to check my formulation.

Another problem is, in the literature the nonlinear response of cornea only becomes obvious for large deflections under very big loads. In my case the pressure is very small (1/100th of the load people apply) and the deflection is small (in the range of several microns) as well. This is puzzling me a lot, and made me to think either my measurement is wrong or other researchers are overlooking an significant fact.

Thank you so much Artur for your response, I will keep searching while keeping your suggestion in mind.

All the best,

Imran

Dear Imran,

you shouldn't try to link the coefficients of polynomial with physical properties, it probably doesn't make sense. Just try to find the simply model for best fit.

If your deflection is only a few micrometers, the nonlinearity probably has completely different source. It's not a matter of geometry or material, but contact stiffness of cornea. On such small distance, probably proper contact (if there is a contact) is not fully developed yet and in fact you maybe are not measuring the stiffness of cornea itself.

For such measurement, main influence on results has measurement equipment for measure force and displacement (especially displacement) and procedure of measurement.

Artur

Dear Arthur,

Thanks a lot once more. I am applying sound wave from a loud speaker to the cornea from a distance and measure the SPL using a microphone. The speaker response is linear. I double checked it. The displacement amplitude is measured by an OCT system, which is specifically designed for these type of experiments. So the measurement system and the method both look fine. I have repeated this experiment more than 10 times and each time got the same nonlinear response. Either I have to find a theoretical model to support our findings or double check every single detail to prove that it is wrong.

Thanks a lot for your comments, I am quite new in this field and I appreciate your help.

Best,

Imran

In 1983 my friend Carlos Luciano Cavalcanti de Oliveira derived the equations you need. Unfortunately I believe he only published the derivations in Portuguese. A copy of the paper can be seen at...

http://www.rbeb.org.br/articles/view/id/4f4fe2bcb1d8f1c52e000000

... from where you can download the pdf.

He relates deflection vs. load, approximations for infinitesimal deflections, and then went on to derive the (linear) relationship between differential capacitance and voltage (he was using one of the plates as one side of a capacitor in a circuit to measure differential pressure).

I hope this helps. The equations are easy to follow even if you cannot read Portuguese ;-)

Best wishes

Fernando

Dear Imran,

your problem became more and more complicated and interesting:)

OCT is ok, question is, if you have high enough time resolution in comparison to dynamic response of a cornea (time to max deflection).

But loudspeaker gives you transient load. In such case, even if you know more or less about dynamic pressure on loaded surface of cornea, this is a dynamic response problem involving inertial forces - and this is another possible source on nonlinear response.

This aspect can be easily checked by relatively simple dynamic numerical models (with FEM). It can be used to check so called frequency response - influence of loading frequency on amplitude of deflection for given load.

It can be done as well experimentally by changing the frequency of loading (if you are using sine load) or shape of impulse (for impulsive load).

Best regards

Artur

Dear Fernando,

Thank you very much for the link, I will definitely check the article you shared. I dont know Portuguese but I have some friends who knows:-)

Best wishes,

Imran

Dear Artur,

In the OCT experiments I am using sinusoidal wave and measuring the corneal deflection over a certain time period, then averaging the collected data. In this sense I believe we have enough time resolution. Regarding your different frequency response suggestion, I also checked that at 3 different frequencies and still load-deflection curve is non-linear. Very recently I used an artificial membrane (made it out of a lab glove) and tested its response which was close to linear. My PI thinks that it could be real but both of us dont know the theory very well so it is hard to explain it.

thanks a lot for your comment again,

Best wishes

Imran

The answer from the bio-mechanics finite element modeling group will include the following aspects:

1) Material is non-linear - in the direction of Mooney–Rivlin, Neo-Hookean or Ogden Material. The simplest case is non-compressible Mooney-Rivlin which requires only one constant. (many references from Ogden)

2) FE Model should include large strains possibility for the shell finite element model.

3) Known models with 1) and 2) are showing good correlations with biomechanical experiments - for validation of material parameters, however, it is better to use the standard 1D tension test - only a fiber. And then to validate the shell model of the real structure (in your case bovine cornea). The final shape of the deflection curve is without any doubts nonlinear.