Elasticity and Fluid Flow in Human Eyes Alistair Fitt School of Mathematics University of...

Elasticity and Fluid Flow in Human Eyes

Alistair Fitt

School of Mathematics

University of Southampton

[email protected]

http://www.maths.soton.ac.uk © Al’s slides 2006

University of Portsmouth 24th April 2006

mailto:[email protected]

Philosophy and collaborators……

• On my soapbox: This talk is about Mathematical Medicine: a “popular area” but should be done not only in a University Department but also with proper medics

• Go to operations! Smell the blood! Dish the gore! etc. etc.

Collaborators: •Rich Braun (University of Delaware)

•Chris Canning (Southampton Eye Unit)

•Jeff Dewynne (Sydney Beach Australia)

•Gabriela Gonzalez (University of Bradford)

© Al’s slides 2006

THE MAIN BITS OF A HUMAN EYE


A range of problems…….

• We have studied quite a wide range of problems concerning fluid flow, elasticity etc. in a human eye

• I only have time to mention a few today

• General philosophy: I will only talk about problems that are currently in progress

• There will not be any “complete stories”

• Remarks that begin “it’s obvious, all you have to do is…….” are FINE as we really want to solve these problems!


Anterior segment flow• It is well known that the aqueous humour in the anterior segment

flows under a number of conditions

• Flow can be caused by:


Buoyancy (external/blood T-difference)

Lens wobble (phakodenesis)

Aqueous production (ciliary body)

Saccadic eye movement

Rapid eye movement (REM)

• Things can also be complicated by the presence of PARTICLES in the eye: these might be red blood cells, white blood cells, or pigment particles

RED BLOOD CELLS (HYPHEMA)


WHITE CELLS (HYPOPYON)


PIGMENT (KRUKENBERG SPINDLE)


We have been able to analyze all of these flows, essentially by using

lubrication theory (anterior chamber is long and thin)

– some examples (not much time for explanations):


BASIC BUOYANCY-DRIVEN FLOW

Flow driven by buoyancy in anterior chamber (external temperature gradient)

)(

12)()(

),(2

)(1)(

)(2

1

)(2

1))(2)((

622

101

033

010

2

201

20

TTh

zTT

wPhPh

yxPTT

gaxpp

HORRIDw

zhzpv

zhzphzzhTTh

zzzhgu

yyxx

a

y

x

SOME OTHER RELATED FLOWS

Gravity/buoyancy driven flow (asleep, face-up) – weak torodial flow

RH half of anterior chamber: secretory flow with no gravity from behind lens to drainage angle via trabecular meshwork


SECRETORY FLOW BALANCES GRAVITY


A truly 3-D flow whose velocity can be written down in closed form – something of a rarity

Both form and topology of streamlines are very complicated: fluid particles “spun out” to drainage angle from deep within centre of anterior chamber

POAG from blockages in the Trabecular Meshwork/Schlemm’s Canal

Primary Open Angle Glaucoma (POAG) is essentially caused by a build

up of pressure due to insufficient outflow

We are just beginning to pose sensible models for this and can make

IOP predictions using couple Friedenwald/lubrication theory models


0 xxxxxxxxt hh

All sorts of silliness is possible!

t (hrs)

pilocarpine

15mmHg

30mmHg

TM blockage

Summary of anterior chamber flows

• Anterior chamber flows are interesting and their analysis can be useful, e.g.

Cold patches for drug distribution

Disruption and cure of Hyphema, Hypopyon etc.

Phakodenesis for Marfan’s syndrome

Analysis of trabecular meshwork blockages


• It is also nice to be able to provide theoretical confirmation of what have up to now

been largely anecdotal observations

• Notwithstanding this, anterior chamber flow is much less of a real issue than

RETINAL DETACHMENT – one of the largest causes of blindness worldwide

MATHEMATICS OF RETINAL DETACHMENT

• Many people suffer from a DETACHED RETINA

• If not treated, this will lead to blindness

• The retina is the “BAG” inside the eye that holds all of the light-sensitive cells.

• When it detaches, vision is badly affected

• Many forms of treatment are now available

- laser spot weld, cryopexy

- bubble insertion, scleral buckling


RETINAL DETACHMENT

Q

AS THE RETINA PEELS AWAY, FLUID ENTERS AND MAKES THE TEAR WORSE.

RESULT:

THE POSTERIOR SEGMENT OF THE EYE IS FILLED WITH FLUID (“VITREOUS HUMOR”). NORMALLY THIS CANNOT FLOW, BUT AFTER “VITREOUS REVERSAL” (PVD) FLOW MAY BE POSSIBLE


A TORN RETINA SEVERE VISION EFFECTS

- RAPID PERMANENT BLINDNESS


Retinal detachment (1) (fluids)

• We need a model of detachment: NOTE – highly speculative!

• Virtually nothing known about the mechanics of the process itself

• Model will tell us:

- how fast?

- how much?

- role of vitreous reversal (PVD)?

- can it stop naturally?

- how does fluid entering affect details?

- how does traction from vitreous affect details?


xL

z

h(x,t)RETINA


choroidO

FLUID ENTERING

h0

VITREOUS TRACTION

Assume h0/L = ε << 1, retina has “surface tension” σ


phULptULt

wLwuUuzLzxLx

)/(,)/(

,,,,20

Equations:

velocity q = uex + wez, pressure p

density ρ, dynamic viscosity μ

vitreous tractional force B = Bez (B > 0)

Usual scalings:

0.

)).(( 2

q

Bqpqqqt


axx

xt

z

z

phtxhxp

txhtxhxutxhtxhxw

txhxu

xw

xutxh

xu

)),(,(

),,()),(,(),()),(,(

,0)),(,(

,0)0,(

),0,(),(3

)0,(

We end up to leading order with familiar equations

Values ε < 1/10, ε2Re 2.5 x 10-4

0

,01

zx

zzzx

wu

Bpup

Boundary conditions:

NO SHEAR AT RETINA

NO SLIP AT CHOROID

GREENSPAN SLIP AT CHOROID

RETINA HAS “SURFACE TENSION”

KINEMATIC B/C


063

3

x

xxxxt

hhhh

Bh

Equations are easily solved to give:

As usual we can also find w etc.

Now integrate from z = 0 to z = h in the usual way.

This gives an evolution equation for h(x,t) in the form

32

222

zhzh

Bhu xxx

x

Many boundary conditions are possible, but for simplicity we first consider the case where no fluid enters the detachment (probably not realistic – but have to start somewhere: most conclusions will carry over)


Need 4 space conditions, one time condition and presumably one for the free boundary

)(

0

3/1

0

),(

)))((()),((

0)),((

0),0(

0),0(

)()0,(

tx

tccx

c

xxx

x

c

Cdxtxh

txKttxh

ttxh

th

th

xHxh

This is just one possible problem prescription

- Before we go on, what about stability?

Initial shape known

Cavity volume conserved

Tanner’s law

Detachment ends at x = xc(t)

Flat where no fluid enters

Zero flux enters


The obvious linearised stability problem is

And of course if we do the usual h = Aei(ωt+kx)

Then it’s sort of obvious that there will be instability

For all k with k2 < B/(σμ)

For “pushing” (B<0) this is fine – but the vitreous “pulls”

Presumably since the instabilities exist only for small wave numbers they are, in reality, too long ever to be seen

0 xxxxxxt hB

hh


063

3

x

xxxxt

hhhh

Bh

Before trying to solve, let’s do some asymptotics:

This will reveal a rather well-known difficulty: for

we analyse the behaviour near the reattachment by trying

))(())(( txxxtxh cn

c

when = 0 the balance gives n = 1 which FAILS – there is no balance that operates to give a detachment that “attaches”

Only for non-zero can the balance operate properly: and then n = 1 is essential for Tanner’s law (if n 1 then the gradient at reattachment is either zero or infinite)


7/1

23

7/17/1

)(

]''''''''['

),(

ttx

fffffKff

xtfth

cc

Now we can get a whole load of solutions:

(1) Similarity:

(2) Numerics:

Transform

Then numerics can all be done using standard central difference: they give predictable results

t

Xtxx c )(

The numerics show solutions spreading out etc., but are not of interest

on their own. Much more interesting are the outstanding questions:


• What about this mystical “surface tension”

• Tanner’s law? Has it any relevance in this case?

• Can we get a handle on how strong the “tractional force” must be?

• Is the retina really stuck down like sellotape?

• What is the real role of the fluid entering?

• Could saccadic motion be important (it looks unlikely)?

• ……….many more questions remain to be answered

RETINAL DETACHMENT (2) (solids)

• A CRAZY QUESTION:• WHAT HAPPENS IF YOU PUT AN ELASTIC BAND AROUND

SOMEONE’S EYE AND TIGHTEN IT?

- In particular, what happens to the focal length of the patient’s eye?

- EVEN MORE IMPORTANT: can tonometer measurements be trusted?

• YEUCCCHHH!• ???!!!!! IS THIS SOME SORT OF TORTURE?


SCLERAL BUCKLE


SCLERAL BUCKLE SURGERY

….fit your own scleral buckle in 6 easy stages…….


SCLERAL BUCKLES

• SCLERAL BUCKLING IS A VERY EFFECTIVE TREATMENT FOR SOME SORTS OF RETINAL DETACHMENT

• The buckle forces the retina back into contact with the choroid

• Note that the buckle normally stays on for life and is not removed

• It also normally causes few problems and is not visible

• ……..BUT WHAT DOES IT DO TO TONOMETRY READINGS?

For patients who have had any sort of eye troubles, nearly all examinations are likely

to begin with a tonometric assessment of the patient’s IOP (intraocular pressure)


TONOMETRY

THE “HAMMER HEAD” IS PRESSED ON TO YOUR EYE UNTIL 3.06mm OF THE CORNEA IS FLATTENED (TOPICAL ANAESTHETIC - DOESN’T HURT!)

THE FORCE NEEDED TO DO THIS IS RELATED TO THE IOP

(1gm force 1mmHg IOP “Imbert-Fick”)

HIGH IOP (SAY IOP > 22mmHg) IS SERIOUS


A GOLDMANN TONOMETER

GLAUCOMA

THE TROUBLE WITH HIGH IOP IS THAT IT IS VERY OFTEN ASSOCIATED WITH

GLAUCOMA (IOP > 25 mmHg, say)

GLAUCOMA IS ACTUALLY OPTIC NERVE NEUROPATHY – ASSOCIATED WITH MANY EYE PROBLEMS

It is thought that high IOP bends the lamina cribrosa affecting the fenestration of the optic nerve and thereby damaging it (can do elasticity calculations!)

In any case, it makes you go blind……….


MATHEMATICS OF TONOMETRY

• Tonometry has been around for over 100 years but for many years nobody really knew how or why it worked.

• Calculations were always based on the “Imbert-Fick” principle, but surgeons have always “made allowances” for thick corneas etc.

• A “black art” where interpretations vary.

• The idea here is to use some simple elasticity calculations to try to put the Imbert-Fick principle on a firmer theoretical footing and also to try to guide us to understand what we have to do for a buckled eye

• Note that similar theory could also be used to understand tonometry for eyes that have undergone LASIK sight correction


r = a

r = b

TONOMETER

BUCKLE


Governing equations and boundary conditionsThe equations that have to be solved are the linear elasticity equations

(assuming axisymmetry)

Fr

Fr

rrr

rrrrrrrr

]cot)(3[

]cot2[

,,

,,


Where rr. r, and denote the stresses in the sclera.

Fr and F are the restoring forces that equilibrate the tonometer force

The Boundary conditions are

0)()(

)(

)()(

ba

IOPpb

Fa

rr

atmrr

rr

It turns out that it is possible to solve this problem using a “Love Stress function”

. This has the properties that

0sinsin

122

22

24

rrrr

rrrrr

rrrrr

rrrEEu

rr

r

sincos

sincos)1(

sincos

sincos)2(

sincos

1cos

)1(2

2

22

2

22

22

with similarly horrid expressions for u, σr, σ and the Love function satisfies


We can now use the spherically symmetric eigenfunctions of the biharmonic

equation to write

0

)(cos)(n

nn PfF

0

211 )(cos)(),(n

nn

nn

nn

nn

n PrDrCrBrAr

Where the Pn(cos) are standard Legendre polynomials and the coefficients An, Bn, Cn

and Dn are to be determined.

Things now get a bit ugly and even using MAPLE some carefully-planned

calculations are required if things are not to get out of hand. We now write


We can now find fairly easily that

)(1)1(

2

1

4

)21(

2

1

8

11

4

31

32

45

10

3

432432

3232

nn

On

f

n

fD

nO

n

f

n

fC

nO

n

f

n

fB

nO

n

f

n

fA

nnn

nnn

nnn

nnn

)1)((28

5,

)1)((10

)(3

)23)((24

5,

)12)((4

)(3

551

4

233

330

1

551

54

2330

33

1

ab

faD

ab

IOPbafD

ab

fbaB

ab

IOPfbaB

but the general coefficients satisfy linear equations that are very large and unwieldy.

Rather surprisingly, these can actually be solved in closed form (no infinite equations)

The expressions are too horrible to display, but crucially we can show fairly easily that

(convergence assured) © Al’s slides 2006

We can now actually do the problems that we have set out to address:

TONOMETRY:

))(cos)(cos(2

)1(cos2

11

0

nnn

atm

PPT

f

Tpf

where for a standard Goldmann tonometer with an eye globe of external radius 12.5mm

The elastic constants have all been measured: υ = 0.3, E = 9000000 Pa,

a = 0.0125 m, b = 0.01198, patm = 101080, α = /25 etc.

NOTE: in the results that follow, there are NO “fiddle factors”!


2α 2/25.6

“Normal” eye with IOP = 20mmHg


BUCKLED EYE (“EXPERIMENT” REFERS TO THEORY)

SHOWS THAT AS IOP INCREASES STANDARD IMBERT-FICK LAW MAY BECOME UNRELIABLE


Non-applanation problems at larger IOPs


AREA

W/IOP

IMBERT-FICK

IOP = 30mmHg

IOP = 15mmHG

IOP = 20mmHg + BUCKLE

Flattened area vs. tonometer force/IOP ratio (differences slightly exaggerated)


MIXED BVP

It turns out that to do the tonometry and scleral buckle problems properly, the

equation (for f())

0

0

)cos,2()cos,2()(),(

sin)(),()(

n

nPnPnqH

where

dfHg

Must be solved. This is a FREDHOLM 1st KIND integral equation, and as such is very ill-posed


END RESULT

• The whole problem can be solved

• The Imbert-Fick principle may be verified

• We can see when the principle is wrong

• We can see how to allow for corneal thickness

• We can predict post-buckling focal length

• We can give adjustments for buckled eyes


AND I DIDN’T EVEN HAVE TIME TO TELL YOU ABOUT…..

• CATARACTS, FALSE LENSES AND EJECTOR SEATS

• EYES AND ELECTRICITY• BLINKING AND DRY EYE• DRUGS IN THE EYE• HOW EYES REPAIR THEMSELVES

• ETC…..ETC…..


Elasticity and Fluid Flow in Human Eyes Alistair Fitt School of Mathematics University of...

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