Collagen-Proteoglycan Structural Interactions in

Corneal Biomechanics: Fundamental Mechanics and Computational Modeling

Green Light Meeting

Xi Cheng Advisor: Peter M. Pinsky

Department of Mechanical Engineering Stanford University

The human cornea

• Unique tissue (strong + transparent)

• Primary optical component of eye

• Surgical corrections modify the cornea

• Diseased/pathological conditions require biophysical

description / explanation

• Goal: comprehensive model of cornea based on first principle

Current modeling methodology

rayxlamellastroma WW

volF WWW lamella

• Incompressible elastic solid

• Valid for short time scale

• Cannot predict swelling and hydration change

• Cannot distinguish ex vivo/in vivo

• Clear need for new modeling methodology

In vivo cornea

• Electrolyte gel reinforced by collagen fibers

• Active ionic transport modulates osmotic pressure and hydration

• Transport of metabolic species may modify corneal hydration

• No existent 3-D model for gel-collagen interaction

P = 0

P = 15 mmHg

P = ?

active ion

transport

metabolic

species

passive fluid

transport

Main questions

• How to model swelling and hydration?

• How to model the effect of active ionic pumping on hydration?

• How does collagen architecture influence swelling and the stability of cornea?

• What is the role of proteoglycans in transparency?

• What are the roles of metabolic species?

Modeling electrolyte gel

• Triphasic Model (Lai et al. 1998)

– Model the solid displacement, fluid pressure and ionic concentrations as independent

fields

– The fully-coupled system is difficult to solve in general situations

• Energy approach

– Equivalent to the triphasic theory under thermodynamic equilibrium

– Well-suited to model collagen-swelling interactions

– Extended to include active transport for living cornea

– First time for the cornea, we model the interplay of actively modulated osmotic

pressure, fluid pressure and stromal collagen fiber elasticity in an energy framework.

CCACAC ELMFL WWWW )(),(),(

fibril matrix

Free energy

density electrolyte

Non-equilibrium

Equilibrium

thermodynamic equilibrium

Illustration of depth-dependent swelling

Volume Dilation J

1

3

Free Swelling

In vivo swelling 42.9

-182.2

uz (μm)

Fibril lattice self-organization

Electrostatic forces only

Entropic elastic forces only

Thesis Outline

Chapter 1. Introduction

1.1 Significance of the corneal swelling

1.2 Proteoglycans in the corneal stroma

1.3 Stroma collagen architecture

Chapter 2. A thermodynamic approach of modeling the stromal swelling pressure

2.1 Background

2.2 Electrostatic free energy

2.3 Entropic elastic free energy

2.4 Donnan based approximation

2.5 Results

Chapter 3. Modeling the endothelial active ion transport

3.1 Background

3.2 Kedem-Katchalsky theory

3.3 Modified electrostatic free energy

3.4 Analytical approximation for osmotic pressure

3.5 Model calibration from imbibition pressure

Thesis Outline

Chapter 4. A macroscopic model for hydration and collagen-swelling interaction in

the in vivo human

4.1 Background

4.2 Collagen organization and stromal elasticity

4.3 Approximation of the modified electrostatic free energy

4.4 Results

4.4.1 Confined and unconfined swelling pressure

4.4.2 Collagen-swelling interaction

4.4.3 Swelling of a cornea with Fuch’s dystrophy

4.4.4 Swelling due to changes in intraocular pressure

Chapter 5. Mechanisms of self-organization for the collagen fibril lattice in the

human cornea

5.1 Background

5.2 Interfibrillar lattice restoring forces

5.3 Lattice dynamics model with an imperfect GAGs topology

5.4 Results

Thesis Outline

Chapter 6. Modeling of corneal metabolism and swelling

6.1 Background

6.2 Conservation law

6.3 Reaction model

6.4 Results

Chapter 7. Discussion

7.1 Effect of the inclined lamellae in maintaining the stability of the refractive surface

7.2 Effect of the GAGs in maintaining the fibril lattice organization

7.3 Potential future model extensions