 
                  THE FIRST LETTER FROM ABROAD

     I like to travel abroad, down narrow country roads lined
with fruit trees, bounded by fields of grain that sway gently in
the breezes of the late summer.  I am especially fond of the
small university towns with their half-timbered houses clustered
around the medieval market place.  Visits to such destinations
become somewhat more affordable when I can combine them with a
professional purpose; then the cost of my travels becomes
deductible on my Federal Income Tax return; for which reason I
have made it a habit to make a friendly courtesy call to the
local optometrist or ophthalmologist whose office is convenient
to my itinerary.  I talk with them about glaucoma, the only topic
about which I am able to display even a veneer of knowledge.
Thus I obtain some idea of how these eye problems are treated in
other parts of the world, and sometimes, it pleases me, they want
to know how we diagnose and treat glaucoma in Cambridge.  Of
course one meets many different personalities on such junkets,
which are made all the more interesting by their unpredictability. 
One such imaginary encounter, which stands out in my memory,
is the subject of this month's Glaucoma Letter.  

     On this occasion, I found myself in a German speaking
district.  I was particularly eager to meet with an oculist
because several days had gone by since the last such encounter,
and I understood that my expenses might be disallowed unless such
a meeting took place soon.  I had been directed to what in our
country would be an old apartment building, constructed perhaps a
hundred years ago.  Its outside was covered with a rather
monotonous brown stucco, and impatient students had cut a path
across the narrow ribbon of lawn that bordered it.  Upstairs were
apartments or dormitory rooms obviously occupied by students.
The windows were open, and one could hear that within a not
unaccomplished oboist was practicing his scales.  The downstairs
rooms had been converted into offices, one of which belonged to a
professor of ophthalmology at the local medical school with whom
I had made an appointment.  The door to the professor's office
was open.  The waiting room was furnished with simple maroon-
covered armchairs which were empty.  The secretary, a tall lady
with greying hair turned at her desk to welcome me.  She had a
pleasantly modulated voice.

     "Guten Morgen", she said, "You are no doubt the gentleman
from the Cambridge Glaucoma Foundation who wishes to speak with
Professor K.  He is waiting for you.  Please go straight down the
hall.  You will find him in the last room on your left."

     I did as I was told.  I knocked on the designated door, and
a voice from within bade me enter.  "Guten Tag, setzen sie sich
nur, dort in den Untersuchungsstuhl," and he added with a smile,
"Und ich verspreche auch ihre Augen nicht zu untersuchen."
     "I am sorry, Herr Professor," I began, my German is not that
good.  I must ask you to repeat what you were saying."
     "Oh, my apologies, we can speak English, I was merely
inviting you to sit in my examining chair, and I promised not to
examine your eyes.  Tell me what brings you here."

     I said I was traveling abroad, visiting various optometrists
and ophthalmologists, interested in confirming the similarities
and understanding the differences between the way glaucoma is
diagnosed and treated in our respective cities.  "Ein sehr
interessantes Unternehmen," he began, and then he quickly
corrected himself.  "A fascinating project. But tell me first,
where are you from?"
     "I come from Cambridge, Massachusetts," I began. He looked
perplexed.  "That is not in England?"
     "In the United States, near Boston," I explained.
     "I have never heard of it. Does it have a university?"
     I was speechless, and there ensued a silence which began to
embarrass me. Not so, Professor K. who resumed the conversation.
     "Is is anywhere near St. Louis?" he asked. I shook my head.

     "The optic disc," he said, "What a puzzle. Do you understand
how the excavation occurs?"
     "I wish I did," I said.
     "I suspect we often skip over the details of what we
observe, because we assume that we all agree, and sometimes
because we are afraid to reveal our ignorance."

     "Have you ever observed glaucomatous excavation to regress?"
I began.
     "Yes, I think I have. In infants whose congenital glaucoma
is relieved by goniotomy, a deep excavation may disappear, and
the disc will then actually appear quite normal."
     "What about optic excavation in the adult. Do you think that
can disappear?"
     "It surely does not occur very often, but once every few
years I see a patient where I think that is just what must have
taken place.  As the patient's glaucoma progresses, the
excavation becomes more and more severe, and we become more and
more concerned that he might lose field. We then use increasingly
strong miotics; the pupil becomes very small; the disc impossible
to see. Often cataract develops and inspite of our best efforts,
field loss supervenes."
     "But surely under those circumstances, you do not expect the
disc excavation to regress."
     "No, not just then. Such a patient may require filtering
surgery; it is performed, the pressure becomes very low, the
cataract progresses.  Some months or years later, the cataract
may be removed, usually by means of a cataract extraction from
below, and then when the wound has healed and the media have
become clear, when one looks, one may be surprised, it is the
exception, I repeat, not the rule, to find that the disc is not
so badly cupped after all, indeed that it is quite flat."
     "What about its color?"
     "I should have mentioned that. The color is pale.  The disc
is atrophic."
     "And the field?"
     "The field is usually badly damaged, so we are sure that
there is something amiss with the eye, but we wonder whether it
was really glaucoma."
     "Rather than what?"
     "Oh, some vascular problem, something like anterior ischemic
neuropathy, for example. We do not really know.  Since the pupil
was miotic all the while that the excavation was progressing, we
had no chance to photograph it, and without a photograph, we
wonder if we might not have made a mistake in our description."

     "I make mistakes all the time," I said.

     "Isn't it strange how our preconceptions of what we will
find affect our thinking, if not indeed our observation. If the
disc had indeed been cupped, we would have though nothing of it,
but when our observations conflict with our preconceptions, we
would sooner deny what we see than modify what we believe.
Perhaps that is one way we perpetuate errors from year to year
and from generation to generation."
     "I take it then that in your experience the disc damaged by
elevated intraocular pressure never recovers to become normal."
     "I think not."
     "Does it always get worse?"
     "If the pressure remains elevated, given time, these discs
always deteriorate until there is total cupping."
     "And if the pressure is normal?"
     "The concept of a normal pressure isn't strictly applicable
in these cases. A pressure which would be statistically normal
and tolerated indefinitely by a healthy disc can still cause
progession of the cupping and lead to field loss where the disc
has once been damaged."
     "But if one lowers the pressure sufficiently . . . "
     "It is hard ever to be confident that the pressure is low
enough.  As a practical matter, we get the pressure as low as
possible with medication, or with surgery if it is appropriate,
and hope for the best."

     "Can you give me some idea of how rapidly excavation of the
nervehead develops?"
     "That depends on the pressure. The higher the pressure, the
faster the progression. You pose a mathematical issue and we
should try to respond to it in mathematical terms.  The rate of
excavation appears to be a function of the pressure.  It is also
a function of the time. Neither function, however, is linear."
     "What do you mean when you say a function is not linear."
     "Simply, that when one variable is plotted against the other,
one obtains a straight line. First, the rate of progression is
not a fixed proportion of the pressure; The assumption that
subjected to a pressure of 36, a disc deteriorates one and a half
times as rapidly as it might at a pressure of 24, is not correct.
The rate of excavation at 36 mm. Hg. is not one and a half, but
perhaps seven times as great as the rate of excavation at 24 mm.
Hg.  Second, the extent of progression is not a constant multiple
of the time.  If the rate of excavation were linear with time,
then a disc subjected to a pressure of 36 for two years would
sustain twice as much excavation as a disc subjected to such a
pressure for one year.  This also is not the case.  The rate of
excavation accelerates, and after the second year, the damage is
significantly greater than twice the damage that might have been
observed after the first year."
     "I conceive of the relationship between the pressure and the
rate of excavation of the disc as a geometric curve.  It is a
curve which, I believe, rises very rapidly, exponentially in
fact. I have no reason to think that it is not everywhere
continuous. I believe it has a derivative at every point."

     I didn't know what to say. I groped for an appropriate
reply.
     "It is a fascinating disease," I began, "Provided you don't
have it yourself," Professor K. completed my sentence.
     Incidentally, I have a tentative explanation of the factors
that affect the rate of excavation. Would you like to hear it?"
     "Very much so," I said, "if I am not taking too much of your
time."
     "It is not the time that concerns me," he said solemnly, "It
is the potential misunderstanding. Please remember that I said my
explanation was very much tentative."
     "I would still like to hear it," I said.
     "Before I continue, tell me whether my account of the
relationship of time and pressure to the rate of disc excavation
corresponds to what doctors in your country understand of
glaucoma."
     "I myself agree," I said. "I would rather not speak for the
others."  "You are very cautious," Professor K. said, nodding his
head as if he assented, I don't know to what.

     All the while we were speaking, a large green spider had
been making its way along a strip of moulding high above our
heads.  Now the spider appeared to have reached its destination,
for it paused, and began to lower itself, at first slowly, then
ever faster, by means of a thread which it was spinning as it
descended.  Professor K. must have noticed my distracted gaze.
He turned, and when he noticed the spider, nodded to it as if it
were a not unwelcome friend.  "He is much better at this sort of
thing than I am," he said.
     The shadow of perplexity suffused his face.  "Can you help
me," he said. "I need help.  Is it politically incorrect to
refer to the spider as "he"?  Will the women consider themselves
slighted if I refer to the spider as "he", or will they consider
themselves maligned if I refer to the spider as "she"?  I wish to
offend no one.  The German language makes it easy for us.  It
leaves us no discretion in the choice of gender.  Die Spinne, the
spider, is feminine even when we refer to the Don Juan of
arachnoids."
     "Don't worry about my sensibilities, Herr Professor," I
said, "Whatever you call them, I make sure I stay free of their
nets."
     "Well spoken," he said, and lapsed into silence.  We watched
the spider begin to build its web.  "That web ill be complete
long before mine, and much more craftsmanlike."  There ensued
another lengthy pause.

     "I have a mathematical model for the process of glaucomatous
excavation of the optic disc," began Professor K. diffidently, as
if he were revealing some clandestine illicit undertaking.
"Please tell me about it," I asked.  "But first," he replied
deliberately, "I must extract from you a promise.  Will you give
it to me?"  "Tell me what I must promise."  "Promise me that you
will never treat a patient by a mathematical formula."  "I
promise," I said, but as I spoke, the absurdity of the promise
came home to me.  Either the man was a fool, or he was making a
fool of me.  Perhaps, I mused, he is exercising a peculiar sense
of humor.  Professor K. accepted my promise solemnly.  His
features betrayed no trace of a smile.

     You asked how rapidly excavation of the nerve develops.  To
begin with, remember that all the figures that we are able to
cite have only statistical validity.  Consider that subject to
the same intraocular pressure, one eye may deteriorate to a given
level of visual loss in five years, another in four, six or
seven.  Not only will any "average" value that we hypothesize
reflect intuitive judgment rather than empirical measurements,
but even if the measurements were feasible, the averages
calculated from them would give us only limited information about
any given eye.  For example, the economists are fond of talking
about the average annual personal income in a given region.  If
we are told that in a city of 1000 persons, the average annual
personal income is 2500 Mark, this might mean that each of the
1000 inhabitants had exactly an income of 2500 Mark; but it might
also mean that of the 1000 inhabitants, one had an income of 2.5
Million Mark, and 999  had no income at all.  This might mean
that each of the 1000 inhabitants had exactly an income of 2500
Mark; but it might also mean that of the 1000 inhabitants, one
had an income of 2.5 Million Mark, and 999  had no income at all.
Nonetheless, statistical values are useful, provided we interpret
them properly and use them sensibly.  If I said that at a
pressure of 36, an eye deteriorates seven times as fast as it
would at a pressure of 24, you must interpret that figure to be
but an average, approximate value, inferred from the model.  I
suspect, and assume, that the actual values are widely scattered,
which is the reason why the course of glaucomatous cupping is so
relatively unpredictable, and why the effects of therapy, for
example, are so difficult to assess.  Yet if we presume to deal
rationally  with the problems of glaucoma diagnosis and therapy,
we have no alternative but to try to understand the distribution
of values with reference to their mean."
     "The mathematical model of which I speak relies on two
fundamental assumptions. Let me articulate them for you, and you
may decide whether they are sufficiently plausible to make the
modeling process worthwhile.  The first of these assumptions is
that the optic disc may be considered a segment of a thin-walled
sphere, and that excavation of the disc, via whatever mechanism,
is a consequence and proportional to, the stresses induced in
that sphere by the intraocular pressure."

     I said nothing.

     "It seems to me not only a plausible, but perhaps an
unavoidable assumption," he continued, "since only a solid
structure can contain the intraocular pressure.  Therefore the
forces induced in the disc and its components may properly be
treated as stresses."
     "You remember then the formula for stresses induced in a
thin-walled sphere is

                       S = p*r/(2*w)

'r' is the radius of the globe.  'p' is the intraocular
pressure.  'w' is the thickness of the disc, and 'S', of course,
is the induced stress.  If, therefore, over a period of time, an
elevated intraocular pressure 'p' brings about a thinning of the
disc, it will effect a decrease in the value of 'w', and a
corresponding increase in the stress, 'S'.  If now we adhere to
our assumption that the destructive effects of the intraocular
pressure are brought about by the stress which that pressure
induces in the disc, then for any given pressure acting over a
period of time, as the disc becomes attenuated, the destructive
effect of that pressure will continuously increase.  Accordingly,
and this is the second assumption, the model specifies that the
pressure multiplied by some appropriate constant equals the rate,
continuously compounded, at which the excavation deepens.
Mathematically this entails the assumption of some slow
fundamental rate of attenuation of the optic disc, R0, which is
multiplied by the exponentiated pressure, exp(p*k).  Thus we
write simply: R1=R0*exp(p*k), where R0 is some constant to be
empirically determined, which corresponds to the "natural" rate
of excavation in the absence of all intraocular pressure, and 'k'
is a second constant which corrects the scale and the dimension
of the pressure. 'R1' is the rate of excavation which we wish to
find.  'p' is the intraocular pressure which we assume to remain
constant.  Finally, if we are interested also in the effect of
differences of disc thickness and globe diameter on the process
of excavation, we may, in the above equation, replace 'p' with
'S' and write:
                         R1=R0*exp(S*k)

or, substituting for 'S, write

                      R1=R0*exp(p*r*k/2*w))
This is our model.

     "I don't understand it," I said.

     "I think I can explain it to you, if you care to spend the
time," Professor K. said apologetically. "It is late now. Perhaps
you would like to come back tomorrow or the day after." My head
had started to ache. I dreaded the thought of coming back. I
glanced at the spider who was making progress with the
construction of her web.  As yet she had caught no fly.  It
seemed obvious that Professor K. was eager to tell someone about
his mathematical model.  Perhaps he could get no one else to
listen.  I made an effort to be polite.  "Thank you," I said as I
rose from the examining chair.  "I will come back if it is at all
possible.  I wish I were a more competent mathematician."


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Copyright 2006, Ernst Jochen Meyer
